This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2014) (preloaded format=pdflatex 2014.5.25) 9 NOV 2015 14:48 entering extended mode restricted \write18 enabled. %&-line parsing enabled. **UDG-Thresholds-Fixed.tex (./UDG-Thresholds-Fixed.tex LaTeX2e <2014/05/01> Babel <3.9k> and hyphenation patterns for 78 languages loaded. ! Undefined control sequence. l.9 \chapter {Second moment method with applications} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.9 \chapter{S econd moment method with applications} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.18 L et us toss a fair coin $n$ times and estimate what is the probability ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! 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Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 18--19 [] [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 n$ [] Overfull \hbox (12.38847pt too wide) in paragraph at lines 18--19 []\OML/cmm/m/it/10 n$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (3.44513pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 i$ [] Overfull \hbox (22.16953pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 X[] \OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 18--19 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (22.16953pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 X[] \OT1/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 18--19 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (19.62494pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (37.29056pt too wide) in paragraph at lines 18--19 [][] 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[] Overfull \hbox (29.33568pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 n=\OT1/cmr/m/n/10 2] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (15.00005pt too wide) in paragraph at lines 18--19 \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 3$ [] Overfull \hbox (36.11232pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X \OMS/c msy/m/n/10 ^^@ [] Overfull \hbox (19.35878pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 ^^V\OMS/cmsy/m/n/10 j ^^U [] Overfull \hbox (60.81136pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (9.32367pt too wide) in paragraph at lines 18--19 []$ [] Overfull \hbox (14.16663pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = [] Overfull \hbox (26.0024pt too wide) in paragraph at lines 18--19 []$ [] Overfull \hbox (36.11232pt too wide) in paragraph at lines 18--19 \OML/cmm/m/it/10 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.20 W e will use the following result for the second moment method. You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 20--21 [] [] ! LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.22 \begin{theorem} \label{thm:2mm} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.23 I f $X\ge 0$ is a r.v. taking integer values, $Pr[X=0]\le \frac{Var[X]}... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.24 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.25 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.29 \end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 23--30 [] [] Overfull \hbox (19.62494pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (35.00113pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (28.74028pt too wide) in paragraph at lines 23--30 []$ [] Overfull \hbox (43.31133pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] = [] Overfull \hbox (39.69447pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/ m/it/10 X\OT1/cmr/m/n/10 ][])$ [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 n$ [] Overfull \hbox (35.00113pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0] = [] Overfull \hbox (17.62505pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (1)$ [] Overfull \hbox (35.00113pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (35.0012pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 [\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X \OMS/c msy/m/n/10 ^^@ [] Overfull \hbox (35.91661pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ]\OMS/cmsy /m/n/10 j ^^U [] Overfull \hbox (25.3611pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ]]$ [] Overfull \hbox (14.16663pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = [] Overfull \hbox (37.07364pt too wide) in paragraph at lines 23--30 []$ [] Overfull \hbox (36.11232pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X \OMS/c msy/m/n/10 ^^@ [] Overfull \hbox (35.91661pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ]\OMS/cmsy /m/n/10 j ^^U [] Overfull \hbox (60.81136pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (9.32367pt too wide) in paragraph at lines 23--30 []$ [] Overfull \hbox (35.00113pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (28.74028pt too wide) in paragraph at lines 23--30 []$ [] Overfull \hbox (43.31133pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] = [] Overfull \hbox (39.69447pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/ m/it/10 X\OT1/cmr/m/n/10 ][])$ [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 n$ [] Overfull \hbox (35.00113pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0] = [] Overfull \hbox (78.46138pt too wide) in paragraph at lines 23--30 [][] [] \OMS/cmsy/m/n/10 ! [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0$ [] Overfull \hbox (35.00113pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 Pr\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 23--30 \OT1/cmr/m/n/10 0] = [] Overfull \hbox (17.62505pt too wide) in paragraph at lines 23--30 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (1)$ [] ! LaTeX Error: The font size command \normalsize is not defined: there is probably something wrong with the class file. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.30 Your command was ignored. Type I to replace it with another command, or to continue without it. [1 ! Undefined control sequence. \newlabel{chap:2MM}{{}{\thepage }} l.30 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{chap:2MM}{{}{}} {/usr/local/texlive/2014/texmf-var/fonts/map/pdftex/updmap/pdftex.map} ! Undefined control sequence. \newlabel{thm:2mm}{{}{\thepage }} l.30 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{thm:2mm}{{}{}} ] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.31 W e will see applications of this theorem in this chapter. You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 31--32 [] [] ! Undefined control sequence. l.33 \section {Cliques in a random graph} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.33 \section{C liques in a random graph} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no C in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! ! LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.35 \begin{theorem} \label{thm:clique-2mm} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.39 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.41 T o prove the first part of this theorem, we will estimate how many subs... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.45 N ext we show the second part of Theorem \ref{thm:clique-2mm} using the ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! 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Undefined control sequence. ...ence `thm:clique-2mm' on page \thepage \space undefined\on@line . l.45 ...econd part of Theorem \ref{thm:clique-2mm} using the Second Moment M... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `thm:clique-2mm' on page undefined on input line 45. Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no S in font nullfont! 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Missing character: There is no m in font nullfont! ! Undefined control sequence. ...: Reference `thm:2mm' on page \thepage \space undefined\on@line . l.48 ...[X]^2)$ and then use Theorem \ref{thm:2mm} to conclude that the prob... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `thm:2mm' on page undefined on input line 48. Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! 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Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Overfull \hbox (37.62495pt too wide) in paragraph at lines 49--51 $\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ][] = [] Overfull \hbox (55.72214pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ]\OML/cmm/ m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] = [] Overfull \hbox (165.52141pt too wide) in paragraph at lines 49--51 [][] \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ] [ ][] \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 k$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 S$ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 T$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 S$ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 T$ [] Overfull \hbox (16.15279pt too wide) in paragraph at lines 49--51 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 S \OMS/cmsy/m/n/10 \ [] Overfull \hbox (20.56595pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 T\OMS/cmsy/m/n/10 j \OML/cmm/m/it/10 < [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 49--51 \OT1/cmr/m/n/10 2$ [] Overfull \hbox (52.73158pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/10 ] = [] Overfull \hbox (55.68996pt too wide) in paragraph at lines 49--51 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ]\OML/cm m/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (237.78651pt too wide) detected at line 53 []\OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] = [] [] \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/10 ] \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\O T1/cmr/m/n/10 ]\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/ m/n/10 ] [] Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ...ence `eqn:clique-var' on page \thepage \space undefined\on@line . l.54 ...X_T]$. Thus, Equation \ref{eqn:clique-var} can be rewritten as The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `eqn:clique-var' on page undefined on input line 54. Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Overfull \hbox (6.70831pt too wide) in paragraph at lines 53--55 $\OML/cmm/m/it/10 S$ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 53--55 \OML/cmm/m/it/10 T$ [] Overfull \hbox (16.15279pt too wide) in paragraph at lines 53--55 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 S \OMS/cmsy/m/n/10 \ [] Overfull \hbox (20.56595pt too wide) in paragraph at lines 53--55 \OML/cmm/m/it/10 T\OMS/cmsy/m/n/10 j ^^U [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 53--55 \OT1/cmr/m/n/10 2$ [] Overfull \hbox (42.17607pt too wide) in paragraph at lines 53--55 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (55.68996pt too wide) in paragraph at lines 53--55 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ]\OML/cm m/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 53--55 [] [] Overfull \hbox (288.9904pt too wide) detected at line 57 []\OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] = [] [] [] \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/1 0 ] \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[ ]\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/c mr/m/n/10 ]\OML/cmm/m/it/10 : [] Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ...ce `eqn:clique-var-2' on page \thepage \space undefined\on@line . l.58 ...ality from Equation \ref{eqn:clique-var-2} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `eqn:clique-var-2' on page undefined on input line 58 . Missing character: There is no . in font nullfont! Overfull \hbox (55.68996pt too wide) in paragraph at lines 57--59 $\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ]\OML/ cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 57--59 [] [] Overfull \hbox (225.52264pt too wide) detected at line 61 []\OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] \OMS /cmsy/m/n/10 ^^T [] [] [] \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X []X[]\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.63 N ow observe that there are $n\choose k$ possibilities for choosing vert... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! 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Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 63--64 [] [] Overfull \hbox (14.11005pt too wide) in paragraph at lines 63--64 []$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 63--64 \OML/cmm/m/it/10 S$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 63--64 \OML/cmm/m/it/10 k$ [] Overfull \hbox (13.58688pt too wide) in paragraph at lines 63--64 []$ [] Overfull \hbox (3.18057pt too wide) in paragraph at lines 63--64 \OML/cmm/m/it/10 l$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 63--64 \OML/cmm/m/it/10 S$ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 63--64 \OML/cmm/m/it/10 T$ [] Overfull \hbox (24.78023pt too wide) in paragraph at lines 63--64 []$ [] Overfull \hbox (12.23267pt too wide) in paragraph at lines 63--64 \OML/cmm/m/it/10 T \OMS/cmsy/m/n/10 n [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 63--64 \OML/cmm/m/it/10 S$ [] Overfull \hbox (186.58939pt too wide) detected at line 66 []\OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] \OMS /cmsy/m/n/10 ^^T [] [][][]\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X []X[]\OT1/cmr/m/n/10 ]\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.68 L et us compute $E[X_SX_T]$. Since $X_S$ and $X_T$ are 0-1 r.v., $E[X_SX... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 68--69 [] [] Overfull \hbox (42.17607pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (14.0778pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (14.58438pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (42.17607pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/10 ]$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 S$ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 T$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 S$ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 T$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 k$ [] Overfull \hbox (6.70831pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 S$ [] Overfull \hbox (13.29865pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (3.18057pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 l$ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 T$ [] Overfull \hbox (13.375pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 S \OMS/cmsy/m/n/10 [ [] Overfull \hbox (7.23265pt too wide) in paragraph at lines 68--69 \OML/cmm/m/it/10 T$ [] Overfull \hbox (26.3647pt too wide) in paragraph at lines 68--69 \OT1/cmr/m/n/10 2[] \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (13.15285pt too wide) in paragraph at lines 68--69 []$ [] Overfull \hbox (96.47545pt too wide) detected at line 71 []\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/10 ] = [][] [] Missing character: There is no S in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ... `eqn:clique-var-sum' on page \thepage \space undefined\on@line . l.72 ...X_T]$ in Equation \ref{eqn:clique-var-sum} , we obtain The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `eqn:clique-var-sum' on page undefined on input line 72. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Overfull \hbox (42.17607pt too wide) in paragraph at lines 71--73 $\OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X[]X[]\OT1/cmr/m/n/10 ] $ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 71--73 [] [] Overfull \hbox (182.97948pt too wide) detected at line 75 []\OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] \OMS /cmsy/m/n/10 ^^T [] [][][]\OT1/cmr/m/n/10 2[]\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.77 N ext we show that $Var[X]/E[X]^2$ is $o(1)$. Note that $E[X]={n \choose... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no D in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 77--79 [] [] Overfull \hbox (64.82527pt too wide) in paragraph at lines 77--79 \OML/cmm/m/it/10 Var\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ]\OML/cm m/m/it/10 =E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ][]$ [] Overfull \hbox (17.62505pt too wide) in paragraph at lines 77--79 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (1)$ [] Overfull \hbox (33.13882pt too wide) in paragraph at lines 77--79 \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 ] = [] Overfull \hbox (37.50945pt too wide) in paragraph at lines 77--79 []\OT1/cmr/m/n/10 2[]$ [] Overfull \hbox (130.58218pt too wide) detected at line 79 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 l\OT1/cmr/m/n/10 ) = []\OML /cmm/m/it/10 : [] Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Overfull \hbox (90.4894pt too wide) detected at line 80 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 l\OT1/cmr/m/n/10 ) = []\OML /cmm/m/it/10 : [] Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (84.30855pt too wide) detected at line 84 [][] \OMS/cmsy/m/n/10 ^^T [] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/1 0 l\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.86 C onsider You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! ! LaTeX Error: Environment align* undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.87 \begin{align*} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no f in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. l.88 f(2)& =\frac{ {k \choose 2} {{n-k}\choose {k-2}} {2^{2\choose 2}} } {{n... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no = in font nullfont! Missing character: There is no k in font nullfont! ! Missing $ inserted. $ l.88 ... {k-2}} {2^{2\choose 2}} } {{n \choose k}} \\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten $. {k \choose 2} {{n-k}\choose {k-2}} {2^{2\choose 2}} l.88 ... {k-2}} {2^{2\choose 2}} } {{n \choose k}} \\ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Missing $ inserted. $ l.88 ... {k-2}} {2^{2\choose 2}} } {{n \choose k}} \\ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing } inserted. } l.88 ... {k-2}} {2^{2\choose 2}} } {{n \choose k}} \\ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. $ l.88 ... {k-2}} {2^{2\choose 2}} } {{n \choose k}} \\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \over #2} l.88 ... {k-2}} {2^{2\choose 2}} } {{n \choose k}} \\ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Misplaced alignment tab character &. & l.89 & =\frac{ {k(k-1)} {{n-k}\choose {k-2}}} {{n \choose k}} I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align*}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.90 \end{align*} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.90 \end{align*} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing } inserted. } l.90 \end{align*} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Overfull \hbox (20.0pt too wide) in paragraph at lines 86--91 [] [] Overfull \hbox (116.85991pt too wide) in paragraph at lines 86--91 []$$[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.92 S ince $k=(2+\epsilon)\log n< to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...ence `thm:clique-2mm' on page \thepage \space undefined\on@line . l.92 ... the proof of Theorem \ref{thm:clique-2mm} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `thm:clique-2mm' on page undefined on input line 92. Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 92--93 [] [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 k \OT1/cmr/m/n/10 = [] Overfull \hbox (16.66672pt too wide) in paragraph at lines 92--93 \OT1/cmr/m/n/10 (2 + [] Overfull \hbox (48.53354pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 ^^O\OT1/cmr/m/n/10 ) [] \OML/cmm/m/it/10 n << [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 n$ [] Overfull \hbox (29.30559pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (2) = [] Overfull \hbox (17.62505pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (1)$ [] Overfull \hbox (29.82642pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k\OT1/cmr/m/n/10 ) = [] Overfull \hbox (17.62505pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (1)$ [] Overfull \hbox (27.48615pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 l\OT1/cmr/m/n/10 ) = [] Overfull \hbox (17.62505pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (1)$ [] Overfull \hbox (13.73608pt too wide) in paragraph at lines 92--93 \OML/cmm/m/it/10 l \OT1/cmr/m/n/10 = [] Overfull \hbox (32.74292pt too wide) in paragraph at lines 92--93 \OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; [] ; k$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 92--93 [] [] ! Undefined control sequence. l.96 \section {Thresholds for Random Geometric Graphs} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.96 \section{T hresholds for Random Geometric Graphs} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.104 A graph property $\mathcal{P}$ is {\em increasing} if a graph $G$ sati... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.106 B y~\cite{goel2004sharp} any monotone property of a random geometric gr... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no [ in font nullfont! \citation{goel2004sharp} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.106 By~\cite{goel2004sharp} any monotone property of a random geometric gr... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `goel2004sharp' on page undefined on input line 106. 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Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no [ in font nullfont! \citation{Gupta1998} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.108 ...998,Panchapakesan2001,penrose1997longest} . Here we investigate thre... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Gupta1998' on page undefined on input line 108. Missing character: There is no , in font nullfont! \citation{Panchapakesan2001} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.108 ...998,Panchapakesan2001,penrose1997longest} . Here we investigate thre... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Panchapakesan2001' on page undefined on input line 10 8. Missing character: There is no , in font nullfont! \citation{penrose1997longest} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.108 ...998,Panchapakesan2001,penrose1997longest} . Here we investigate thre... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `penrose1997longest' on page undefined on input line 1 08. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (25.43053pt too wide) in paragraph at lines 106--109 [] [] [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 r$ [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 106--109 \OT1/cmr/m/n/10 1$ [] Overfull \hbox (18.78008pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ! [] Overfull \hbox (10.00002pt too wide) in paragraph at lines 106--109 \OMS/cmsy/m/n/10 1$ [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 106--109 \OMS/cmsy/m/n/10 P$ [] Overfull \hbox (17.39127pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 t\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 )$ [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 r \OT1/cmr/m/n/10 = [] Overfull \hbox (30.0163pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 n\OT1/cmr/m/n/10 ))$ [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 106--109 \OMS/cmsy/m/n/10 P$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 G\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\OT1/cmr/m/n/10 )$ [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 r \OT1/cmr/m/n/10 = [] Overfull \hbox (31.75238pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 !\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 n\OT1/cmr/m/n/10 ))$ [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 106--109 \OMS/cmsy/m/n/10 P$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 G\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\OT1/cmr/m/n/10 )$ [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 106--109 \OMS/cmsy/m/n/10 P$ [] Overfull \hbox (17.39127pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 t\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 )$ [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 r \OT1/cmr/m/n/10 = [] Overfull \hbox (30.0163pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 o\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 n\OT1/cmr/m/n/10 ))$ [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 106--109 \OMS/cmsy/m/n/10 P$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 G\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\OT1/cmr/m/n/10 )$ [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 r \OT1/cmr/m/n/10 = [] Overfull \hbox (31.75238pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 !\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 (\OML/cmm/ m/it/10 n\OT1/cmr/m/n/10 ))$ [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 106--109 \OMS/cmsy/m/n/10 P$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 G\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\OT1/cmr/m/n/10 )$ [] Overfull \hbox (37.00471pt too wide) in paragraph at lines 106--109 []$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 106--109 \OML/cmm/m/it/10 G\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\OT1/cmr/m/n/10 )$ [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 106--109 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 106--109 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 106--109 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.112 \noindent {\bf Related Work: } You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. l.112 \noindent {\bf Related Work: } The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no : in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{gilbert1959random} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.114 ...ed by Gilbert in \cite{gilbert1959random} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `gilbert1959random' on page undefined on input line 11 4. Missing character: There is no ] in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! LaTeX Font Info: Try loading font information for +cmr on input line 115. LaTeX Font Info: No file cmr.fd. on input line 115. LaTeX Font Warning: Font shape `/cmr/m/n' undefined (Font) using `/cmr/m/n' instead on input line 115. ! Corrupted NFSS tables. wrong@fontshape ...message {Corrupted NFSS tables} error@fontshape else let f... l.115 and Erd\"{o} s and R\'{e}nyi \cite{erd6s1960evolution}. This error message was generated by an \errmessage command, so I can't give any explicit help. Pretend that you're Hercule Poirot: Examine all clues, and deduce the truth by order and method. LaTeX Font Warning: Font shape `/cmr/m/n' undefined (Font) using `OT1/cmr/m/n' instead on input line 115. Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no [ in font nullfont! \citation{erd6s1960evolution} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.115 ...s and R\'{e}nyi \cite{erd6s1960evolution} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `erd6s1960evolution' on page undefined on input line 1 15. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{gilbert1961random} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.118 in \cite{gilbert1961random} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `gilbert1961random' on page undefined on input line 11 8. Missing character: There is no ] in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (5.43053pt too wide) in paragraph at lines 112--128 [] [] Overfull \hbox (10.00003pt too wide) in paragraph at lines 112--128 \/cmr/m/n/10 ^^?o [] Overfull \hbox (4.44444pt too wide) in paragraph at lines 112--128 \/cmr/m/n/10 ^^Se [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 112--128 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 112--128 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.129 A n instance of Erd\"{o}s-R\'{e}nyi graph~\cite{erd6s1960evolution} is ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no [ in font nullfont! \citation{erd6s1960evolution} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.129 ...R\'{e}nyi graph~\cite{erd6s1960evolution} is obtained by taking $n$... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `erd6s1960evolution' on page undefined on input line 1 29. Missing character: There is no ] in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no [ in font nullfont! \citation{bollobas2001random} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.129 ...iedgut1996every,goel2004sharp,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `bollobas2001random' on page undefined on input line 1 29. Missing character: There is no , in font nullfont! \citation{Bradonjic2014} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.129 ...iedgut1996every,goel2004sharp,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Bradonjic2014' on page undefined on input line 129. Missing character: There is no , in font nullfont! \citation{friedgut1996every} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.129 ...iedgut1996every,goel2004sharp,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `friedgut1996every' on page undefined on input line 12 9. Missing character: There is no , in font nullfont! \citation{goel2004sharp} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.129 ...iedgut1996every,goel2004sharp,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `goel2004sharp' on page undefined on input line 129. Missing character: There is no , in font nullfont! \citation{Mccolm2001} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.129 ...iedgut1996every,goel2004sharp,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Mccolm2001' on page undefined on input line 129. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no H in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 129--131 [] [] Overfull \hbox (9.44446pt too wide) in paragraph at lines 129--131 \/cmr/m/n/10 ^^?o^^Se [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 129--131 [] [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 n$ [] Overfull \hbox (5.03125pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 p$ [] Overfull \hbox (19.80316pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 G[]$ [] Overfull \hbox (19.80316pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 G[]$ [] Overfull \hbox (5.03125pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 p$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 r$ [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 129--131 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 129--131 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 129--131 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 129--131 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 129--131 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (9.44446pt too wide) in paragraph at lines 129--131 \/cmr/m/n/10 ^^?o^^Se [] Overfull \hbox (19.80316pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 G[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 129--131 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.133 B ollob\'{a}s and Thomason~\cite{Bollobas1987} showed that You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no B in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{Bollobas1987} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.133 ...ob\'{a}s and Thomason~\cite{Bollobas1987} showed that The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Bollobas1987' on page undefined on input line 133. Missing character: There is no ] in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no [ in font nullfont! \citation{friedgut1996every} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.134 ...iedgut and Kalai~\cite{friedgut1996every} , and a result of Bourgain... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `friedgut1996every' on page undefined on input line 13 4. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no [ in font nullfont! \citation{Bourgain1998} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.134 ...of Bourgain and Kalai~\cite{Bourgain1998} . In the Erd\"{o}s-R\'{e}n... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Bourgain1998' on page undefined on input line 134. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no [ in font nullfont! \citation{alon2004probabilistic} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.135 ...p= 1/n$; see~\cite{alon2004probabilistic} . The planarity threshold ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `alon2004probabilistic' on page undefined on input lin e 135. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no [ in font nullfont! \citation{bollobas2001random} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.135 ...~\cite{bollobas2001random,spencer1987ten} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `bollobas2001random' on page undefined on input line 1 35. Missing character: There is no , in font nullfont! \citation{spencer1987ten} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.135 ...~\cite{bollobas2001random,spencer1987ten} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `spencer1987ten' on page undefined on input line 135. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (25.00002pt too wide) in paragraph at lines 133--136 []\/cmr/m/n/10 ^^Sa [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 133--136 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 133--136 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 133--136 [] [] Overfull \hbox (9.44446pt too wide) in paragraph at lines 133--136 \/cmr/m/n/10 ^^?o^^Se [] Overfull \hbox (19.80316pt too wide) in paragraph at lines 133--136 \OML/cmm/m/it/10 G[]$ [] Overfull \hbox (15.58676pt too wide) in paragraph at lines 133--136 \OML/cmm/m/it/10 p \/cmr/m/n/10 = [] Overfull \hbox (31.58803pt too wide) in paragraph at lines 133--136 [] \OML/cmm/m/it/10 n=n$ [] Overfull \hbox (15.58676pt too wide) in paragraph at lines 133--136 \OML/cmm/m/it/10 p \/cmr/m/n/10 = [] Overfull \hbox (16.00238pt too wide) in paragraph at lines 133--136 \/cmr/m/n/10 1\OML/cmm/m/it/10 =n$ [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 133--136 [] [] Overfull \hbox (19.80316pt too wide) in paragraph at lines 133--136 \OML/cmm/m/it/10 G[]$ [] Overfull \hbox (15.58676pt too wide) in paragraph at lines 133--136 \OML/cmm/m/it/10 p \/cmr/m/n/10 = [] Overfull \hbox (16.00238pt too wide) in paragraph at lines 133--136 \/cmr/m/n/10 1\OML/cmm/m/it/10 =n$ [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 133--136 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 133--136 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.137 A general reference on random geometric graphs is You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no [ in font nullfont! \citation{penrose2003random} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.138 \cite{penrose2003random} . There is extensive literature on various asp... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `penrose2003random' on page undefined on input line 13 8. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no [ in font nullfont! \citation{hall1985coverage} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.140 \cite{hall1985coverage,janson1986random} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `hall1985coverage' on page undefined on input line 140 . Missing character: There is no , in font nullfont! \citation{janson1986random} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.140 \cite{hall1985coverage,janson1986random} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `janson1986random' on page undefined on input line 140 . Missing character: There is no ] in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no [ in font nullfont! \citation{balister2008percolation} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.141 ...louring by~\cite{balister2008percolation} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `balister2008percolation' on page undefined on input l ine 141. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{Bradonjic2014} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.142 ...l2004sharp,Krishnamachari2002,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Bradonjic2014' on page undefined on input line 142. Missing character: There is no , in font nullfont! \citation{goel2004sharp} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.142 ...l2004sharp,Krishnamachari2002,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `goel2004sharp' on page undefined on input line 142. Missing character: There is no , in font nullfont! \citation{Krishnamachari2002} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.142 ...l2004sharp,Krishnamachari2002,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Krishnamachari2002' on page undefined on input line 1 42. Missing character: There is no , in font nullfont! \citation{Mccolm2001} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.142 ...l2004sharp,Krishnamachari2002,Mccolm2001} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Mccolm2001' on page undefined on input line 142. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 137--143 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.144 R andom geometric graphs have a connectivity threshold of $\sqrt{\ln n/... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no R in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no [ in font nullfont! \citation{Gupta1998} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.144 ...998,Panchapakesan2001,penrose1997longest} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Gupta1998' on page undefined on input line 144. Missing character: There is no , in font nullfont! \citation{Panchapakesan2001} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.144 ...998,Panchapakesan2001,penrose1997longest} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Panchapakesan2001' on page undefined on input line 14 4. Missing character: There is no , in font nullfont! \citation{penrose1997longest} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.144 ...998,Panchapakesan2001,penrose1997longest} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `penrose1997longest' on page undefined on input line 1 44. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no [ in font nullfont! \citation{Gupta1998} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.145 Gupta and Kumar~\cite{Gupta1998} provided a connectivity threshold for... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Gupta1998' on page undefined on input line 145. Missing character: There is no ] in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no [ in font nullfont! \citation{Penrose1999} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.146 By a result of Penrose~\cite{Penrose1999} , in $\dg$, any threshold fun... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Penrose1999' on page undefined on input line 146. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no [ in font nullfont! \citation{Panchapakesan2001} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.146 ...an and Manjunath~\cite{Panchapakesan2001} showed that $\sqrt{\ln n/... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Panchapakesan2001' on page undefined on input line 14 6. Missing character: There is no ] in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no [ in font nullfont! \citation{Penrose1999} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.150 ...d in~\cite{Penrose1999,penrose2003random} . Connectivity of random g... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Penrose1999' on page undefined on input line 150. Missing character: There is no , in font nullfont! \citation{penrose2003random} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.150 ...d in~\cite{Penrose1999,penrose2003random} . Connectivity of random g... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `penrose2003random' on page undefined on input line 15 0. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no J in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no [ in font nullfont! \citation{Godehardt1996} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.150 ...dehardt and Jaworski~\cite{Godehardt1996} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Godehardt1996' on page undefined on input line 150. Missing character: There is no ] in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no [ in font nullfont! \citation{Appel2002} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.151 Appel and Russo~\cite{Appel2002} considered the connectivity under the... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `Appel2002' on page undefined on input line 151. Missing character: There is no ] in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no . in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.154 \noindent {\bf Organization: } You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. l.154 \noindent {\bf Organization: } The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no O in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no a in font nullfont! 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If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `connected-section' on page undefined on input line 1 56. Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! 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Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no a in font nullfont! 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Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ...rence `plane-section' on page \thepage \space undefined\on@line . l.156 ...or planar. In Section~\ref{plane-section} , we prove that $n^{-2/3}$... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `plane-section' on page undefined on input line 156. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ...ence `planar-section' on page \thepage \space undefined\on@line . l.156 ...be plane. In Section~\ref{planar-section} , we prove that $n^{-5/8}$... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `planar-section' on page undefined on input line 156. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! 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Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! 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Overfull \hbox (10.86105pt too wide) in paragraph at lines 154--160 [] [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 154--160 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 154--160 \OML/cmm/m/it/10 k$ [] Overfull \hbox (24.9963pt too wide) in paragraph at lines 154--160 \OML/cmm/m/it/10 n[]$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 154--160 \OML/cmm/m/it/10 k$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 154--160 [] [] Overfull \hbox (24.82185pt too wide) in paragraph at lines 154--160 \OML/cmm/m/it/10 n[]$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 154--160 [] [] Overfull \hbox (24.82185pt too wide) in paragraph at lines 154--160 \OML/cmm/m/it/10 n[]$ [] ! Undefined control sequence. l.163 \subsection {The threshold for having a connected subgraph on $k$ points} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.163 \subsection{T he threshold for having a connected subgraph on $k$ points} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! 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Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no k in font nullfont! 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Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no ` in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! 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LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.167 \begin{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.169 L et $k\ge 2$ be an integer constant. Then, $n^{\frac{-k}{2k-2}}$ is a ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.170 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.171 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no L in font nullfont! 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Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! 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Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 169--173 [] [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 169--173 \/cmr/m/n/10 2$ [] Overfull \hbox (24.9963pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 n[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 k$ [] Overfull \hbox (52.23026pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 P[]; [] ; P[]$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 k$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 169--173 \OML/cmm/m/it/10 X[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.174 \[ X_i = You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Overfull \hbox (20.0pt too wide) in paragraph at lines 174--174 [][] [] ! Misplaced alignment tab character &. l.176 1 & \quad \text{if } \idg{P_i}\text{ is connected,}\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.176 1 & \quad \text {if } \idg{P_i}\text{ is connected,}\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.176 ...1 & \quad \text{if } \idg{P_i}\text { is connected,}\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. l.177 0 & \quad \text{otherwise.}\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.177 0 & \quad \text {otherwise.}\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (186.82881pt too wide) detected at line 179 \OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] []\OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML /cmm/m/it/10 P[]\/cmr/m/n/10 ][][] 0 [][] [] Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (9.06943pt too wide) in paragraph at lines 179--181 $\OML/cmm/m/it/10 X$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 179--181 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 179--181 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.182 \begin{equation} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Overfull \hbox (20.0pt too wide) in paragraph at lines 182--182 [] [] Overfull \hbox (52.90555pt too wide) detected at line 185 []\OML/cmm/m/it/10 X \/cmr/m/n/10 = [] \OML/cmm/m/it/10 X[]: [] Missing character: There is no O in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (34.53067pt too wide) in paragraph at lines 185--187 $\/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]\/cmr/m/n/10 ] = [] Overfull \hbox (35.66956pt too wide) in paragraph at lines 185--187 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 185--187 \/cmr/m/n/10 1]$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 185--187 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (83.39822pt too wide) detected at line 190 []\/cmr/m/n/10 E[\OML/cmm/m/it/10 X\/cmr/m/n/10 ] = []E[\OML/cmm/m/it/10 X[]\/c mr/m/n/10 ]\OML/cmm/m/it/10 : [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.192 W e obtain an upper bound and a lower bound for $\Pr[X_i=1]$. First, pa... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. ...g: Reference `sS-fig' on page \thepage \space undefined\on@line . l.192 ...ft bottom corner; see Figure~\ref{sS-fig} (a). $S_t$ contains at mos... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `sS-fig' on page undefined on input line 192. Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.194 $ $\mbox{if }\idg{P_i}\mbox{ is connected, then } (A_{i,1} \lor A_{i,2}... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Overfull \hbox (20.0pt too wide) in paragraph at lines 194--194 [] [] Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! 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Overfull \hbox (137.70149pt too wide) detected at line 201 []\/cmr/m/n/10 (\OML/cmm/m/it/10 B[] \OMS/cmsy/m/n/10 _ \OML/cmm/m/it/10 B[] \O MS/cmsy/m/n/10 _ [] _ \OML/cmm/m/it/10 B[]\/cmr/m/n/10 )[]\OML/cmm/m/it/10 DG\/ cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ][]\OML/cmm/m/it/10 ; [] Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (234.55876pt too wide) detected at line 205 [][]\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = 1] \OMS/cmsy/m/n/10 ^^U [ ] []\/cmr/m/n/10 [\OML/cmm/m/it/10 B[]\/cmr/m/n/10 ] = [] [][] = []\OML/cmm/m/i t/10 r[]: [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.207 S ince $k\ge 2$ is a constant, Inequalities~\eqref{Xi-upper} and~\eqref... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! 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Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! ! Undefined control sequence. l.207 ...\ge 2$ is a constant, Inequalities~\eqref {Xi-upper} and~\eqref{Xi-l... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no X in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! ! Undefined control sequence. l.207 ... Inequalities~\eqref{Xi-upper} and~\eqref {Xi-lower} and Equation~\e... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no X in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.207 ... and~\eqref{Xi-lower} and Equation~\eqref {E-X-Xi} imply that The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.208 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: \mathrm allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.209 \Ex [X_i]&= \label{E-Xi} \Theta(r^{2k-2}),\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no E in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no X in font nullfont! ! Missing $ inserted. $ l.209 \Ex[X_ i]&= \label{E-Xi} \Theta(r^{2k-2}),\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.209 \Ex[X_i]& = \label{E-Xi} \Theta(r^{2k-2}),\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced alignment tab character &. l.210 \Ex[X] & =\label{E-X}\Theta(n^kr^{2k-2}). I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.211 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.211 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.213 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.214 \Pr [X\ge 1] &\leq \notag \Ex[X] \mbox{ (by Markov's Inequality)}\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.214 \Pr [X\ge 1] & \leq \notag \Ex[X] \mbox{ (by Markov's Inequality)}\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.214 \Pr [X\ge 1] &\leq \notag \Ex[X] \mbox{ (by Markov's Inequality)}\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. & l.215 & = \notag \Theta(n^k r^{2k-2}) \mbox{ (by~\eqref{E-X})} \\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.215 &= \notag \Theta(n^k r^{2k-2}) \mbox{ (by~\eqref{E-X})} \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. (by~\eqref {E-X}) l.215 ...ta(n^k r^{2k-2}) \mbox{ (by~\eqref{E-X})} \\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. & l.216 & = \label{1mom2:eq} o(1). I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.217 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.217 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 207--219 [] [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 207--219 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 207--219 \/cmr/m/n/10 2$ [] Overfull \hbox (16.66258pt too wide) in paragraph at lines 207--219 []\/cmr/m/n/10 ] = [] Overfull \hbox (46.70953pt too wide) in paragraph at lines 207--219 [] \/cmr/m/n/10 ^^B(\OML/cmm/m/it/10 r[]\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] [] Overfull \hbox (31.98608pt too wide) in paragraph at lines 207--219 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X\/cmr/m/n/10 ] = [] Overfull \hbox (55.96542pt too wide) in paragraph at lines 207--219 [] \/cmr/m/n/10 ^^B(\OML/cmm/m/it/10 n[]r[]\/cmr/m/n/10 )\OML/cmm/m/it/10 :$ [] Overfull \hbox (18.78008pt too wide) in paragraph at lines 207--219 \OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ! [] Overfull \hbox (10.00002pt too wide) in paragraph at lines 207--219 \OMS/cmsy/m/n/10 1$ [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 207--219 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (37.62134pt too wide) in paragraph at lines 207--219 \OML/cmm/m/it/10 o\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (33.12497pt too wide) in paragraph at lines 207--219 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 207--219 \/cmr/m/n/10 1] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (21.43057pt too wide) in paragraph at lines 207--219 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X\/cmr/m/n/10 ][][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 207--219 \/cmr/m/n/10 = [] Overfull \hbox (50.40993pt too wide) in paragraph at lines 207--219 \/cmr/m/n/10 ^^B(\OML/cmm/m/it/10 n[]r[]\/cmr/m/n/10 )[][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 207--219 \/cmr/m/n/10 = [] Overfull \hbox (23.18054pt too wide) in paragraph at lines 207--219 [] \OML/cmm/m/it/10 o\/cmr/m/n/10 (1)\OML/cmm/m/it/10 :$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 207--219 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 207--219 \OML/cmm/m/it/10 k$ [] ! LaTeX Error: Environment figure undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.220 \begin{figure} [htb] Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.220 \begin{figure}[ htb] You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no [ in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ] in font nullfont! ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. \@sharp ->\centering \includegraphics [width=.24\columnwidth ]{2MM/FIG/sS.pdf} l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no [ in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no = in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no 4 in font nullfont! ! Missing number, treated as zero. ] l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). ] l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) Missing character: There is no ] in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no / in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no / in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! ! Missing \endgroup inserted. \endgroup l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! You can't use `macro parameter character #' in restricted horizontal mode. \@preamble ...colsep {\hfil \ignorespaces \@sharp \unskip \hfil }\hskip \tab... l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} Sorry, but I'm not programmed to handle this case; I'll just pretend that you didn't ask for it. If you're in the wrong mode, you might be able to return to the right one by typing `I}' or `I$' or `I\par'. ! Missing } inserted. } l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing } inserted. } l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing \cr inserted. \cr l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I'm guessing that you meant to end an alignment here. ! Missing { inserted. { l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I've put in what seems to be necessary to fix the current column of the current alignment. Try to go on, since this might almost work. ! Missing } inserted. } l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. $ l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing } inserted. } l.224 ...s[width=.24\columnwidth]{2MM/FIG/sS.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Misplaced alignment tab character &. l.225 & \multicolumn{1}{m{.5\columnwidth}}{\centering\includegraphics[width=.... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced \omit. \multispan ->\omit \@multispan l.225 ...[width=.24\columnwidth]{2MM/FIG/sSx.pdf}} I expect to see \omit only after tab marks or the \cr of an alignment. Proceed, and I'll ignore this case. ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.225 ...[width=.24\columnwidth]{2MM/FIG/sSx.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. ...nextchar \ifcase \@chclass \@classz \or \@classi \or \@classii... l.225 ...[width=.24\columnwidth]{2MM/FIG/sSx.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.225 ...[width=.24\columnwidth]{2MM/FIG/sSx.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. ...nextchar \ifcase \@chclass \@classz \or \@classi \or \@classii... l.225 ...[width=.24\columnwidth]{2MM/FIG/sSx.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \@mkpream ...hclass }\ifcase \@lastchclass \@acol \or \or \@preamerr \@ne \o... l.225 ...[width=.24\columnwidth]{2MM/FIG/sSx.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: There's no line here to end. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.226 \\ (a) & (b) Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.226 \\ (a) & (b) I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (20.0pt too wide) in paragraph at lines 220--226 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.226 \\( a) & (b) You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. l.226 \\(a) & (b) I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced \crcr. \endtabular ->\crcr \egroup \egroup $\egroup l.227 \end{tabular} $ I can't figure out why you would want to use a tab mark or \cr or \span just now. If something like a right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Extra }, or forgotten \endgroup. \endtabular ->\crcr \egroup \egroup $\egroup l.227 \end{tabular} $ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Extra }, or forgotten \endgroup. \endtabular ->\crcr \egroup \egroup $\egroup l.227 \end{tabular} $ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Extra }, or forgotten $. \egroup l.227 \end{tabular} $ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! LaTeX Error: \begin{document} ended by \end{tabular}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.227 \end{tabular} $ Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.227 \end{tabular} $ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! LaTeX Error: \caption outside float. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.228 \caption {(a) The square $S_t$ has $s_t$ on its left bottom corner. (... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Missing } inserted. } l.228 \caption{(a) The square $ S_t$ has $s_t$ on its left bottom corner. (... I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no S in font nullfont! ! Missing $ inserted. $ l.228 \caption{(a) The square $S_ t$ has $s_t$ on its left bottom corner. (... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! ! Extra }, or forgotten \endgroup. l.228 ...square $S_x$ which is centered at $s_x$.} I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! LaTeX Error: \begin{document} ended by \end{figure}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.230 \end{figure} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (64.20154pt too wide) in paragraph at lines 226--231 []$$[]$ [] Overfull \hbox (8.20834pt too wide) in paragraph at lines 226--231 \OML/cmm/m/it/10 s[]$ [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 226--231 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (9.72223pt too wide) in paragraph at lines 226--231 \OML/cmm/m/it/10 s[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.232 I n the rest of the proof, we assume that $r=\omega(n^{\frac{-k}{2k-2}}... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no [ in font nullfont! \citation{alon2004probabilistic} ! Undefined control sequence. ...: Citation `\@citeb ' on page \thepage \space undefined\on@line . l.232 ...oment method~\cite{alon2004probabilistic} , that $\Pr[X=0]\to 0$ as ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Citation `alon2004probabilistic' on page undefined on input lin e 232. Missing character: There is no ] in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 232--236 [] [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 232--236 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (39.35742pt too wide) in paragraph at lines 232--236 \OML/cmm/m/it/10 !\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 232--236 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 232--236 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.43053pt too wide) in paragraph at lines 232--236 [] [] Overfull \hbox (33.12497pt too wide) in paragraph at lines 232--236 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X \/cmr/m/n/10 = [] Overfull \hbox (20.55553pt too wide) in paragraph at lines 232--236 \/cmr/m/n/10 0] \OMS/cmsy/m/n/10 ! [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 232--236 \/cmr/m/n/10 0$ [] Overfull \hbox (18.78008pt too wide) in paragraph at lines 232--236 \OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ! [] Overfull \hbox (10.00002pt too wide) in paragraph at lines 232--236 \OMS/cmsy/m/n/10 1$ [] Overfull \hbox (94.62207pt too wide) detected at line 241 [][]\/cmr/m/n/10 [\OML/cmm/m/it/10 X \/cmr/m/n/10 = 0] \OMS/cmsy/m/n/10 ^^T []\ OML/cmm/m/it/10 : [] Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Overfull \hbox (33.12497pt too wide) in paragraph at lines 241--244 $[]\/cmr/m/n/10 [\OML/cmm/m/it/10 X \/cmr/m/n/10 = [] Overfull \hbox (20.55553pt too wide) in paragraph at lines 241--244 \/cmr/m/n/10 0] \OMS/cmsy/m/n/10 ! [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 241--244 \/cmr/m/n/10 0$ [] Overfull \hbox (58.16382pt too wide) detected at line 247 [][] \OMS/cmsy/m/n/10 ! \/cmr/m/n/10 0\OML/cmm/m/it/10 : [] Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.248 In view of Identity~\eqref {X-Xi} we have The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no X in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (141.48967pt too wide) detected at line 252 [][]\/cmr/m/n/10 (\OML/cmm/m/it/10 X\/cmr/m/n/10 ) = [] [](\OML/cmm/m/it/10 X[] ; X[]\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.256 ...ts,k\}$. Thus, in view of Equation~\eqref {Var-X}, we have The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (64.25116pt too wide) in paragraph at lines 252--257 $[]\/cmr/m/n/10 (\OML/cmm/m/it/10 X[]; X[]\/cmr/m/n/10 ) = [] Overfull \hbox (44.25125pt too wide) in paragraph at lines 252--257 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]X[]\/cmr/m/n/10 ] \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (59.3901pt too wide) in paragraph at lines 252--257 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]\/cmr/m/n/10 ]E[\OML/cmm/m/it/10 X[]\/cmr/m/ n/10 ] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (36.47345pt too wide) in paragraph at lines 252--257 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]X[]\/cmr/m/n/10 ]$ [] Overfull \hbox (19.1939pt too wide) in paragraph at lines 252--257 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 \ [] Overfull \hbox (23.967pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 P[]\OMS/cmsy/m/n/10 j \/cmr/m/n/10 = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 252--257 \/cmr/m/n/10 0$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (32.6087pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (12.49829pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (64.25116pt too wide) in paragraph at lines 252--257 []\/cmr/m/n/10 (\OML/cmm/m/it/10 X[]; X[]\/cmr/m/n/10 ) = [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 252--257 \/cmr/m/n/10 0$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (19.1939pt too wide) in paragraph at lines 252--257 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 \ [] Overfull \hbox (23.967pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 P[]\OMS/cmsy/m/n/10 j \/cmr/m/n/10 = [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 w$ [] Overfull \hbox (16.87265pt too wide) in paragraph at lines 252--257 \OML/cmm/m/it/10 w \OMS/cmsy/m/n/10 2 [] Overfull \hbox (42.74295pt too wide) in paragraph at lines 252--257 \OMS/cmsy/m/n/10 f\/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; k\OMS/cmsy/m/n/10 g$ [] ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.258 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: \mathrm allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.259 \Var (X) &=\notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Cov (X_i, X_j)\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.259 \Var (X) &=\notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Cov (X_i, X_j)\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. l.259 \Var (X) & =\notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Cov (X_i, X_j)\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no = in font nullfont! ! Undefined control sequence. l.259 \Var (X) &=\notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Cov (X_i, X_j)\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Missing $ inserted. $ l.259 \Var (X) &=\notag \sum _{w=1}^k\sum_{|P_i\cap P_j|=w} \Cov (X_i, X_j)\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. & l.260 & \le \label{Var-X-2} \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Ex[X_iX_j]. I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.261 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.261 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Overfull \hbox (20.0pt too wide) in paragraph at lines 259--262 [] [] Overfull \hbox (134.5761pt too wide) in paragraph at lines 259--262 [][] [][] []\/cmr/m/n/10 (\OML/cmm/m/it/10 X[]; X[]\/cmr/m/n/10 )[] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 259--262 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (122.9094pt too wide) in paragraph at lines 259--262 [] [][] [][] \/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]X[]\/cmr/m/n/10 ]\OML/cmm/m/it/ 10 :$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.263 T he computation of $\Ex[X_i,X_j]$ involves some geometric consideratio... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 263--264 [] [] Overfull \hbox (40.91786pt too wide) in paragraph at lines 263--264 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]; X[]\/cmr/m/n/10 ]$ [] Overfull \hbox (11.61401pt too wide) in paragraph at lines 263--264 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (12.49829pt too wide) in paragraph at lines 263--264 \OML/cmm/m/it/10 X[]$ [] Overfull \hbox (24.1123pt too wide) in paragraph at lines 263--264 \OML/cmm/m/it/10 X[]X[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.265 \[ X_iX_j = You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Overfull \hbox (20.0pt too wide) in paragraph at lines 265--265 [][] [] ! Misplaced alignment tab character &. l.267 1 & \quad \text{if both } \idg{P_i}\text{ and }\idg{P_j}\tex... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.267 1 & \quad \text {if both } \idg{P_i}\text{ and }\idg{P_j}\tex... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.267 ... & \quad \text{if both } \idg{P_i}\text { and }\idg{P_j}\text{ are... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.267 ...oth } \idg{P_i}\text{ and }\idg{P_j}\text { are connected,}\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. l.268 0 & \quad \text{otherwise.}\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.268 0 & \quad \text {otherwise.}\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Overfull \hbox (273.53938pt too wide) detected at line 270 \OML/cmm/m/it/10 X[]X[] \/cmr/m/n/10 = [] []\OML/cmm/m/it/10 DG\/cmr/m/n/10 [\ OML/cmm/m/it/10 P[]\/cmr/m/n/10 ][]\OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/ it/10 P[]\/cmr/m/n/10 ][][] 0 [][] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.272 B y the definition of the expected value we have You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.273 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: \mathrm allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.274 \Ex [X_iX_j]&=\notag\Pr[X_j=1|X_i=1]\Pr[X_i=1]\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no E in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no X in font nullfont! ! Missing $ inserted. $ l.274 \Ex[X_ iX_j]&=\notag\Pr[X_j=1|X_i=1]\Pr[X_i=1]\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.274 \Ex[X_iX_j]& =\notag\Pr[X_j=1|X_i=1]\Pr[X_i=1]\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.274 \Ex[X_iX_j]&=\notag \Pr[X_j=1|X_i=1]\Pr[X_i=1]\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. & l.275 & =\label{E-XiXj}\Pr[X_j=1|X_i=1]\Ex[X_i]. I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.276 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.276 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Overfull \hbox (20.0pt too wide) in paragraph at lines 272--277 [] [] Overfull \hbox (29.16087pt too wide) in paragraph at lines 272--277 []\OML/cmm/m/it/10 X[]\/cmr/m/n/10 ] = [] Overfull \hbox (36.55383pt too wide) in paragraph at lines 272--277 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (29.94733pt too wide) in paragraph at lines 272--277 \/cmr/m/n/10 1\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (45.11398pt too wide) in paragraph at lines 272--277 \/cmr/m/n/10 1] [][\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 272--277 \/cmr/m/n/10 1][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 272--277 \/cmr/m/n/10 = [] Overfull \hbox (39.33154pt too wide) in paragraph at lines 272--277 [] []\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (29.94733pt too wide) in paragraph at lines 272--277 \/cmr/m/n/10 1\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (34.53075pt too wide) in paragraph at lines 272--277 \/cmr/m/n/10 1]E[\OML/cmm/m/it/10 X[]\/cmr/m/n/10 ]\OML/cmm/m/it/10 :$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.278 B y~\eqref{E-Xi}, $\Ex[X_i]= \Theta(r^{2k-2})$. It remains to compute $... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.278 By~\eqref {E-Xi}, $\Ex[X_i]= \Theta(r^{2k-2})$. It remains to compute $... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. ...g: Reference `sS-fig' on page \thepage \space undefined\on@line . l.280 ...k-1)^2$ squares); see Figure~\ref{sS-fig} (b). The area of $S_x$ is ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `sS-fig' on page undefined on input line 280. Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.281 ...c_wr^{2k-2w}$. In view of Equation~\eqref {E-XiXj}, we have The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (54.53067pt too wide) in paragraph at lines 278--282 [] $\/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]\/cmr/m/n/10 ] = [] Overfull \hbox (39.48741pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 ^^B(\OML/cmm/m/it/10 r[]\/cmr/m/n/10 )$ [] Overfull \hbox (36.55383pt too wide) in paragraph at lines 278--282 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (29.94733pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1]$ [] Overfull \hbox (32.6087pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 k$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (5.71527pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 x$ [] Overfull \hbox (16.4161pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 \ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 r$ [] Overfull \hbox (9.72223pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 s[]$ [] Overfull \hbox (5.71527pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 x$ [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (22.18756pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 (2\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (21.45609pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1)\OML/cmm/m/it/10 r \OMS/cmsy/m/n/10 ^^B [] Overfull \hbox (22.18756pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 (2\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (13.67828pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1)\OML/cmm/m/it/10 r$ [] Overfull \hbox (9.72223pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 s[]$ [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (22.18756pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 (2\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (13.37505pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1)[]$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 r$ [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (22.18756pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 (2\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (13.37505pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1)[]$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 278--282 [] [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (27.57416pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 (2\OML/cmm/m/it/10 kr\/cmr/m/n/10 )[]$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (9.65279pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (24.28249pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 4\OML/cmm/m/it/10 k[]r[]$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 w$ [] Overfull \hbox (32.6087pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (13.29865pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 w$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (32.6087pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (59.75938pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 (4\OML/cmm/m/it/10 k[]r[]\/cmr/m/n/10 )[] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (40.70618pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 c[]r[]$ [] Overfull \hbox (21.35648pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 c[] > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 0$ [] Overfull \hbox (36.55383pt too wide) in paragraph at lines 278--282 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (29.94733pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = [] Overfull \hbox (18.33331pt too wide) in paragraph at lines 278--282 \/cmr/m/n/10 1] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (40.70618pt too wide) in paragraph at lines 278--282 \OML/cmm/m/it/10 c[]r[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.283 \begin{equation} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Overfull \hbox (20.0pt too wide) in paragraph at lines 283--283 [] [] Overfull \hbox (187.05356pt too wide) detected at line 286 []\/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]X[]\/cmr/m/n/10 ] \OMS/cmsy/m/n/10 ^^T \OM L/cmm/m/it/10 c[] \OMS/cmsy/m/n/10 ^^A \OML/cmm/m/it/10 r[] \OMS/cmsy/m/n/10 ^^ A \OML/cmm/m/it/10 r[] \/cmr/m/n/10 = \OML/cmm/m/it/10 c[]r[]; [] Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (21.35648pt too wide) in paragraph at lines 286--288 $\OML/cmm/m/it/10 c[] > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 286--288 \/cmr/m/n/10 0$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.289 S ince $P_i$ and $P_j$ are $k$-tuples which share $w$ points, $|P_i\cup... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.289 ...P_j$. Based on this and Inequality~\eqref {E-XiXj-2}, Inequality~\eq... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.289 ...ality~\eqref{E-XiXj-2}, Inequality~\eqref {Var-X-2} turns out to The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 289--290 [] [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 k$ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 w$ [] Overfull \hbox (19.1939pt too wide) in paragraph at lines 289--290 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 [ [] Overfull \hbox (23.967pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]\OMS/cmsy/m/n/10 j \/cmr/m/n/10 = [] Overfull \hbox (18.29866pt too wide) in paragraph at lines 289--290 \/cmr/m/n/10 2\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 w$ [] Overfull \hbox (29.79643pt too wide) in paragraph at lines 289--290 []$ [] Overfull \hbox (18.29866pt too wide) in paragraph at lines 289--290 \/cmr/m/n/10 2\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 w$ [] Overfull \hbox (16.4161pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 [ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 w$ [] Overfull \hbox (16.4161pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 \ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (13.29865pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 w$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (13.29865pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 w$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (53.17729pt too wide) in paragraph at lines 289--290 []$ [] Overfull \hbox (18.29866pt too wide) in paragraph at lines 289--290 \/cmr/m/n/10 2\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 w$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 289--290 \OML/cmm/m/it/10 P[]$ [] ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.291 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: \mathrm allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.292 \Var (X) & \le \notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Ex[X_iX_j]\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.292 \Var (X) & \le \notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Ex[X_iX_j]\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. l.292 \Var (X) & \le \notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Ex[X_iX_j]\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Missing $ inserted. $ l.292 \Var (X) & \le \notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Ex[X_iX_j]\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Undefined control sequence. l.292 \Var (X) & \le \notag \sum_{w=1}^k\sum_{|P_i\cap P_j|=w} \Ex[X_iX_j]\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. & l.293 & \le \notag \sum_{w=1}^{k} {n \choose {2k-w}}{{2k-w}\choose {w, k-w, k... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.293 &\le \notag \sum_{w=1}^{k} {n \choose {2k-w}}{{2k-w}\choose {w, k-w, k... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. & l.294 & \le\notag \sum_{w=1}^{k} c''_w{n^{2k-w}}r^{4k-2w-2}. I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.294 &\le\notag \sum_{w=1}^{k} c''_w{n^{2k-w}}r^{4k-2w-2}. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.295 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.295 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Overfull \hbox (20.0pt too wide) in paragraph at lines 292--296 [] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 292--296 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (117.3539pt too wide) in paragraph at lines 292--296 [][] [][] \/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]X[]\/cmr/m/n/10 ][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 292--296 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (162.73671pt too wide) in paragraph at lines 292--296 [][] [][]\OML/cmm/m/it/10 c[]r[][] [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 292--296 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (109.67284pt too wide) in paragraph at lines 292--296 [][] \OML/cmm/m/it/10 c[][]r[]:$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.297 f or some constants $c''_w>0$. Consider~\eqref{Var-Ex2} and note that b... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! ! Undefined control sequence. l.297 ...some constants $c''_w>0$. Consider~\eqref {Var-Ex2} and note that by... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.297 ...r~\eqref{Var-Ex2} and note that by~\eqref {E-X}, $\Ex[X]^2\ge c''n^{... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 297--298 [] [] Overfull \hbox (21.35648pt too wide) in paragraph at lines 297--298 \OML/cmm/m/it/10 c[] > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 297--298 \/cmr/m/n/10 0$ [] Overfull \hbox (36.47221pt too wide) in paragraph at lines 297--298 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X\/cmr/m/n/10 ][] \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (48.27913pt too wide) in paragraph at lines 297--298 \OML/cmm/m/it/10 c[]n[]r[]$ [] Overfull \hbox (19.99419pt too wide) in paragraph at lines 297--298 \OML/cmm/m/it/10 c[] > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 297--298 \/cmr/m/n/10 0$ [] ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.299 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: \mathrm allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.300 \frac{\Var (X)}{\Ex[X]^2} & \le \notag\sum_{w=1}^{k} \frac{c''_w{n^{2k... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.300 \frac{\Var (X)}{\Ex[X]^2} & \le \notag\sum_{w=1}^{k} \frac{c''_w{n^{2k... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no ) in font nullfont! ! Missing $ inserted. $ l.300 \frac{\Var (X)}{\Ex[X]^2} & \le \notag\sum_{w=1}^{k} \frac{c''_w{n^{2k... I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Extra }, or forgotten $. \frac #1#2->{\begingroup #1\endgroup \over #2} l.300 \frac{\Var (X)}{\Ex[X]^2} & \le \notag\sum_{w=1}^{k} \frac{c''_w{n^{2k... I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Misplaced alignment tab character &. l.300 \frac{\Var (X)}{\Ex[X]^2} & \le \notag\sum_{w=1}^{k} \frac{c''_w{n^{2k... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.300 \frac{\Var (X)}{\Ex[X]^2} & \le \notag \sum_{w=1}^{k} \frac{c''_w{n^{2k... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. l.301 = \notag \sum_{w=1}^{k} \frac{c''_w}{c''}\cdot\frac{1}{n^{w}r^{2w-2}}\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. & l.302 & = \label{Var-Ex2-2}\frac{c''_1}{c''}\cdot\frac{1}{n^{1}r^{0}}+ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.306 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.306 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing } inserted. } l.306 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Overfull \hbox (361.87885pt too wide) in paragraph at lines 300--307 []$[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.308 S ince $r=\omega(n^{\frac{-k}{2k-2}})$, all terms in~\eqref{Var-Ex2-2} ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.308 ...^{\frac{-k}{2k-2}})$, all terms in~\eqref {Var-Ex2-2} tend to zero. ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. l.308 ...ro. This proves the convergence in~\eqref {Var-Ex2}. Thus, $\Pr[X=0]... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.309 \qed\end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Extra \endgroup. \endgroup l.309 \qed\end{proof} Things are pretty mixed up, but I think the worst is over. Overfull \hbox (20.0pt too wide) in paragraph at lines 308--310 [] [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 308--310 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (39.35742pt too wide) in paragraph at lines 308--310 \OML/cmm/m/it/10 !\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (33.12497pt too wide) in paragraph at lines 308--310 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X \/cmr/m/n/10 = [] Overfull \hbox (20.55553pt too wide) in paragraph at lines 308--310 \/cmr/m/n/10 0] \OMS/cmsy/m/n/10 ! [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 308--310 \/cmr/m/n/10 0$ [] Overfull \hbox (18.78008pt too wide) in paragraph at lines 308--310 \OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ! [] Overfull \hbox (10.00002pt too wide) in paragraph at lines 308--310 \OMS/cmsy/m/n/10 1$ [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 308--310 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (39.35742pt too wide) in paragraph at lines 308--310 \OML/cmm/m/it/10 !\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 308--310 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 308--310 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.69054pt too wide) in paragraph at lines 308--310 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.311 I n the following theorem we show that if $k=O(1)$, then $n^{\frac{-k}{... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! 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Overfull \hbox (20.0pt too wide) in paragraph at lines 311--312 [] [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 311--312 \OML/cmm/m/it/10 k \/cmr/m/n/10 = [] Overfull \hbox (20.68336pt too wide) in paragraph at lines 311--312 \OML/cmm/m/it/10 O\/cmr/m/n/10 (1)$ [] Overfull \hbox (24.9963pt too wide) in paragraph at lines 311--312 \OML/cmm/m/it/10 n[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 311--312 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 311--312 \OML/cmm/m/it/10 k$ [] ! LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.313 \begin{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. 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Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.316 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.317 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...nce `connected-k-thr' on page \thepage \space undefined\on@line . l.318 By Theorem~\ref{connected-k-thr} , if $r=o(n^{\frac{-k}{2k-2}})$, then ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `connected-k-thr' on page undefined on input line 318 . Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! 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Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...nce `connected-k-thr' on page \thepage \space undefined\on@line . l.318 ...he proof of Theorem~\ref{connected-k-thr} , which is based on the se... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `connected-k-thr' on page undefined on input line 318 . Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.321 P artition the unit square into a set $\{s_1,\dots,s_{1/{r^2}}\}$ of sq... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no P in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! 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Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 321--322 [] [] Overfull \hbox (62.49767pt too wide) in paragraph at lines 321--322 \OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 s[]; [] ; s[]\OMS/cmsy/m/n/10 g$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 321--322 \OML/cmm/m/it/10 r$ [] Overfull \hbox (9.65279pt too wide) in paragraph at lines 321--322 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (17.56718pt too wide) in paragraph at lines 321--322 \/cmr/m/n/10 2\OML/cmm/m/it/10 r \OMS/cmsy/m/n/10 ^^B [] Overfull \hbox (9.78938pt too wide) in paragraph at lines 321--322 \/cmr/m/n/10 2\OML/cmm/m/it/10 r$ [] Overfull \hbox (8.20834pt too wide) in paragraph at lines 321--322 \OML/cmm/m/it/10 s[]$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 321--322 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 321--322 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (9.65279pt too wide) in paragraph at lines 321--322 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (13.05551pt too wide) in paragraph at lines 321--322 \OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2 [] Overfull \hbox (56.49763pt too wide) in paragraph at lines 321--322 \OMS/cmsy/m/n/10 f\/cmr/m/n/10 1\OML/cmm/m/it/10 ; [] ; \/cmr/m/n/10 1\OML/cmm/ m/it/10 =[]\OMS/cmsy/m/n/10 g$ [] Overfull \hbox (96.18806pt too wide) detected at line 324 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = 1] \OMS/cmsy/m/n/10 ^^T \/c mr/m/n/10 4[]\OML/cmm/m/it/10 r[]: [] Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (62.49767pt too wide) in paragraph at lines 324--326 $\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 s[]; [] ; s[]\OMS/cmsy/m/n/10 g$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 324--326 \OML/cmm/m/it/10 r$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 324--326 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (8.20834pt too wide) in paragraph at lines 324--326 \OML/cmm/m/it/10 s[]$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 324--326 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (108.8242pt too wide) detected at line 328 []\/cmr/m/n/10 [\OML/cmm/m/it/10 X[] \/cmr/m/n/10 = 1] \OMS/cmsy/m/n/10 ^^U []\ OML/cmm/m/it/10 r[]: [] Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.330 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: \mathrm allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.331 \Ex [X_i]&=\notag \Theta(r^{2k-2}),\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no E in font nullfont! Missing character: There is no [ in font nullfont! Missing character: There is no X in font nullfont! ! Missing $ inserted. $ l.331 \Ex[X_ i]&=\notag \Theta(r^{2k-2}),\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Misplaced alignment tab character &. l.331 \Ex[X_i]& =\notag \Theta(r^{2k-2}),\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.331 \Ex[X_i]&=\notag \Theta(r^{2k-2}),\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. l.332 \Ex[X] & =\notag \Theta(n^kr^{2k-2}). I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.332 \Ex[X] &=\notag \Theta(n^kr^{2k-2}). The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.333 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.333 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Overfull \hbox (16.07635pt too wide) in paragraph at lines 328--334 $\OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 328--334 \/cmr/m/n/10 2$ [] Overfull \hbox (16.66258pt too wide) in paragraph at lines 328--334 []\/cmr/m/n/10 ] = [] Overfull \hbox (43.93182pt too wide) in paragraph at lines 328--334 \/cmr/m/n/10 ^^B(\OML/cmm/m/it/10 r[]\/cmr/m/n/10 )\OML/cmm/m/it/10 ; [] [] Overfull \hbox (31.98608pt too wide) in paragraph at lines 328--334 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X\/cmr/m/n/10 ] = [] Overfull \hbox (53.18771pt too wide) in paragraph at lines 328--334 \/cmr/m/n/10 ^^B(\OML/cmm/m/it/10 n[]r[]\/cmr/m/n/10 )\OML/cmm/m/it/10 :$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.335 I n view of Chebyshev's inequality we need to show that $\frac{\Var (X)... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.335 ...\Var (X)$ from above by Inequality~\eqref {Var-X-2}. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no P in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 335--338 [] [] Overfull \hbox (28.12025pt too wide) in paragraph at lines 335--338 []$ [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 n$ [] Overfull \hbox (32.43059pt too wide) in paragraph at lines 335--338 []\/cmr/m/n/10 (\OML/cmm/m/it/10 X\/cmr/m/n/10 )$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 k$ [] Overfull \hbox (9.74942pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (31.72443pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (19.1939pt too wide) in paragraph at lines 335--338 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 \ [] Overfull \hbox (23.967pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 P[]\OMS/cmsy/m/n/10 j \/cmr/m/n/10 = [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 w$ [] Overfull \hbox (5.71527pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 x$ [] Overfull \hbox (16.4161pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 P[] \OMS/cmsy/m/n/10 \ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 r$ [] Overfull \hbox (9.72223pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 s[]$ [] Overfull \hbox (5.71527pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 x$ [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (17.56718pt too wide) in paragraph at lines 335--338 \/cmr/m/n/10 3\OML/cmm/m/it/10 r \OMS/cmsy/m/n/10 ^^B [] Overfull \hbox (9.78938pt too wide) in paragraph at lines 335--338 \/cmr/m/n/10 3\OML/cmm/m/it/10 r$ [] Overfull \hbox (9.72223pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 s[]$ [] Overfull \hbox (32.6087pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 DG\/cmr/m/n/10 [\OML/cmm/m/it/10 P[]\/cmr/m/n/10 ]$ [] Overfull \hbox (13.29865pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.42825pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 w$ [] Overfull \hbox (10.6337pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 P[]$ [] Overfull \hbox (11.16667pt too wide) in paragraph at lines 335--338 \OML/cmm/m/it/10 S[]$ [] Overfull \hbox (103.52678pt too wide) detected at line 340 \/cmr/m/n/10 E[\OML/cmm/m/it/10 X[]X[]\/cmr/m/n/10 ] \OMS/cmsy/m/n/10 ^^T \OML/ cmm/m/it/10 c[]r[]; [] Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...nce `connected-k-thr' on page \thepage \space undefined\on@line . l.341 ...he proof of Theorem~\ref{connected-k-thr} , we can show that for som... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `connected-k-thr' on page undefined on input line 341 . Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no : in font nullfont! ! LaTeX Error: Environment align undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.342 \begin{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: \mathrm allowed only in math mode. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.343 \Var (X) &\le\notag \sum_{w=1}^{k} c''_w{n^{2k-w}}r^{4k-2w-2},\\ You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no V in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. l.343 \Var (X) & \le\notag \sum_{w=1}^{k} c''_w{n^{2k-w}}r^{4k-2w-2},\\ I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Missing $ inserted. $ l.343 \Var (X) &\le \notag \sum_{w=1}^{k} c''_w{n^{2k-w}}r^{4k-2w-2},\\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. ! Undefined control sequence. l.343 \Var (X) &\le\notag \sum_{w=1}^{k} c''_w{n^{2k-w}}r^{4k-2w-2},\\ The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Misplaced alignment tab character &. l.344 \frac{\Var (X)}{\Ex[X]^2} & \le\notag \sum_{w=1}^{k} \frac{c''_w}{c''}\... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Undefined control sequence. l.344 \frac{\Var (X)}{\Ex[X]^2} &\le\notag \sum_{w=1}^{k} \frac{c''_w}{c''}\... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: \begin{document} ended by \end{align}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.345 \end{align} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.345 \end{align} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.347 \qed\end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...erence `clique-k-thr' on page \thepage \space undefined\on@line . l.348 ...consequence of Theorem~\ref{clique-k-thr} , we have the following co... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `clique-k-thr' on page undefined on input line 348. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (21.35648pt too wide) in paragraph at lines 340--349 $\OML/cmm/m/it/10 c[] > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 340--349 \/cmr/m/n/10 0$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 340--349 [] [] Overfull \hbox (35.23956pt too wide) in paragraph at lines 340--349 \OML/cmm/m/it/10 c[]; c[] > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 340--349 \/cmr/m/n/10 0$ [] Overfull \hbox (7.7778pt too wide) in paragraph at lines 340--349 \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (111.33946pt too wide) in paragraph at lines 340--349 [][] \OML/cmm/m/it/10 c[][]r[]; [] [] Overfull \hbox (38.67577pt too wide) in paragraph at lines 340--349 [] \OMS/cmsy/m/n/10 ^^T [] Overfull \hbox (43.25166pt too wide) in paragraph at lines 340--349 [][] [] \OMS/cmsy/m/n/10 ^^A [] Overfull \hbox (37.6307pt too wide) in paragraph at lines 340--349 []\OML/cmm/m/it/10 :$ [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 340--349 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (39.35742pt too wide) in paragraph at lines 340--349 \OML/cmm/m/it/10 !\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 340--349 \OML/cmm/m/it/10 n$ [] Overfull \hbox (5.69054pt too wide) in paragraph at lines 340--349 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 340--349 [] [] ! LaTeX Error: Environment corollary undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.350 \begin{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.351 $ n^{-1}$ is a threshold for $\dg$ to have an edge, and $n^{-\frac{3}{... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{corollary}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.352 \end{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (36.7385pt too wide) in paragraph at lines 351--353 []$\OML/cmm/m/it/10 n[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 351--353 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (18.55519pt too wide) in paragraph at lines 351--353 \OML/cmm/m/it/10 n[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 351--353 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] ! Undefined control sequence. l.355 \subsection {The threshold for $\dg$ to be plane} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.355 \subsection{T he threshold for $\dg$ to be plane} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 355--357 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 355--357 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.358 I n this section we investigate the You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 358--361 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 358--361 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (37.00471pt too wide) in paragraph at lines 358--361 []$ [] ! LaTeX Error: Environment figure undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.362 \begin{figure} [htb] Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.362 \begin{figure}[ htb] You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no [ in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ] in font nullfont! ! Undefined control sequence. l.364 \includegraphics [width=.25\columnwidth]{2MM/FIG/square1.pdf} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no [ in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no = in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no 5 in font nullfont! ! Missing number, treated as zero. ] l.364 \includegraphics[width=.25\columnwidth] {2MM/FIG/square1.pdf} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). ] l.364 \includegraphics[width=.25\columnwidth] {2MM/FIG/square1.pdf} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) Missing character: There is no ] in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no / in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no / in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 1 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! ! LaTeX Error: \caption outside float. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.365 \caption {An square of diameter $r$ which is partitioned into nine sub-... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{figure}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.367 \end{figure} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 362--368 [] [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 362--368 \OML/cmm/m/it/10 r$ [] ! LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.369 \begin{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.371 I f $r \geq \sqrt{\frac{c \ln n}{n}}$, with $c \geq 36$, then w.h.p. $\... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.372 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. 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Type H for immediate help. ... l.375 I n order to prove that w.h.p. $\dg$ is not plane, we show that w.h.p. ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! 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Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. ...eference `fig:square' on page \thepage \space undefined\on@line . l.375 ...es as depicted in Figure~\ref{fig:square} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. 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Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. ...eference `fig:square' on page \thepage \space undefined\on@line . l.385 ...d$ as depicted in Figure~\ref{fig:square} , it is easy to see that t... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `fig:square' on page undefined on input line 385. Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! 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Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.387 \qed\end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (14.88307pt too wide) in paragraph at lines 383--388 $\OML/cmm/m/it/10 c \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (10.00003pt too wide) in paragraph at lines 383--388 \/cmr/m/n/10 36$ [] Overfull \hbox (12.77782pt too wide) in paragraph at lines 383--388 \/cmr/m/n/10 1 \OMS/cmsy/m/n/10 ^^@ [] Overfull \hbox (7.34332pt too wide) in paragraph at lines 383--388 []$ [] Overfull \hbox (5.28589pt too wide) in paragraph at lines 383--388 \OML/cmm/m/it/10 a$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 383--388 \OML/cmm/m/it/10 b$ [] Overfull \hbox (4.32756pt too wide) in paragraph at lines 383--388 \OML/cmm/m/it/10 c$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 383--388 \OML/cmm/m/it/10 d$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 383--388 [] [] Overfull \hbox (21.79976pt too wide) in paragraph at lines 383--388 \/cmr/m/n/10 (\OML/cmm/m/it/10 a; b\/cmr/m/n/10 )$ [] Overfull \hbox (21.75464pt too wide) in paragraph at lines 383--388 \/cmr/m/n/10 (\OML/cmm/m/it/10 c; d\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 383--388 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.69054pt too wide) in paragraph at lines 383--388 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.389 I n fact, Theorem~\ref{thm3} ensures that w.h.p. there exists a pair of... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...ing: Reference `thm3' on page \thepage \space undefined\on@line . l.389 In fact, Theorem~\ref{thm3} ensures that w.h.p. there exists a pair of... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `thm3' on page undefined on input line 389. Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! 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Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 389--393 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 389--393 [] [] Overfull \hbox (33.44522pt too wide) in paragraph at lines 389--393 \/cmr/m/n/10 []$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 389--393 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 389--393 \OML/cmm/m/it/10 r$ [] Overfull \hbox (18.55519pt too wide) in paragraph at lines 389--393 \OML/cmm/m/it/10 n[]$ [] ! LaTeX Error: Environment lemma undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.394 \begin{lemma} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.396 L et $(a,b)$ and $(c,d)$ be two crossing edges in $\dg$, and let $Q$ be... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! 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Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{lemma}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.397 \end{lemma} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.398 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no R in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. ...ference `fig:square1' on page \thepage \space undefined\on@line . l.399 Refer to Figure~\ref{fig:square1} . The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `fig:square1' on page undefined on input line 399. Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! 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Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.402 \qed\end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 396--403 [] [] Overfull \hbox (21.79976pt too wide) in paragraph at lines 396--403 \/cmr/m/n/10 (\OML/cmm/m/it/10 a; b\/cmr/m/n/10 )$ [] Overfull \hbox (21.75464pt too wide) in paragraph at lines 396--403 \/cmr/m/n/10 (\OML/cmm/m/it/10 c; d\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (5.28589pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 a$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 b$ [] Overfull \hbox (4.32756pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 c$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 d$ [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 396--403 [] [] Overfull \hbox (7.90555pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 Q$ [] Overfull \hbox (21.19176pt too wide) in paragraph at lines 396--403 []\OML/cmm/m/it/10 cad$ [] Overfull \hbox (16.05908pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 ^^Y=\/cmr/m/n/10 2$ [] Overfull \hbox (23.70721pt too wide) in paragraph at lines 396--403 \OMS/cmsy/m/n/10 4\OML/cmm/m/it/10 cad$ [] Overfull \hbox (9.53242pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 cd$ [] Overfull \hbox (25.64351pt too wide) in paragraph at lines 396--403 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 cd\OMS/cmsy/m/n/10 j ^^U [] Overfull \hbox (64.27097pt too wide) in paragraph at lines 396--403 []\OMS/cmsy/m/n/10 fj\OML/cmm/m/it/10 ac\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 ; \O MS/cmsy/m/n/10 j\OML/cmm/m/it/10 ad\OMS/cmsy/m/n/10 jg$ [] Overfull \hbox (25.64351pt too wide) in paragraph at lines 396--403 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 cd\OMS/cmsy/m/n/10 j ^^T [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 r$ [] Overfull \hbox (15.16902pt too wide) in paragraph at lines 396--403 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 ac\OMS/cmsy/m/n/10 j$ [] Overfull \hbox (16.04633pt too wide) in paragraph at lines 396--403 \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 ad\OMS/cmsy/m/n/10 j$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 r$ [] Overfull \hbox (9.61345pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 ac$ [] Overfull \hbox (20.49077pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 ad$\/cmr/m/n/10 | [] Overfull \hbox (10.00002pt too wide) in paragraph at lines 396--403 \/cmr/m/n/10 | [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 396--403 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.69054pt too wide) in paragraph at lines 396--403 [] [] ! LaTeX Error: Environment figure undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.404 \begin{figure} [htb] Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.404 \begin{figure}[ htb] You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no [ in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ] in font nullfont! ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. \@sharp ->\centering \includegraphics [width=.3\columnwidth ]{2MM/FIG/square... l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no [ in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no = in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no 3 in font nullfont! ! Missing number, treated as zero. ] l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} A number should have been here; I inserted `0'. (If you can't figure out why I needed to see a number, look up `weird error' in the index to The TeXbook.) ! Illegal unit of measure (pt inserted). ] l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} Dimensions can be in units of em, ex, in, pt, pc, cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one! I'll assume that you meant to say pt, for printer's points. To recover gracefully from this error, it's best to delete the erroneous units; e.g., type `2' to delete two letters. (See Chapter 27 of The TeXbook.) Missing character: There is no ] in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no / in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no G in font nullfont! Missing character: There is no / in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no 2 in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! ! Missing \endgroup inserted. \endgroup l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! You can't use `macro parameter character #' in restricted horizontal mode. \@preamble ...colsep {\hfil \ignorespaces \@sharp \unskip \hfil }\hskip \tab... l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} Sorry, but I'm not programmed to handle this case; I'll just pretend that you didn't ask for it. If you're in the wrong mode, you might be able to return to the right one by typing `I}' or `I$' or `I\par'. ! Missing } inserted. } l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing } inserted. } l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing \cr inserted. \cr l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I'm guessing that you meant to end an alignment here. ! Missing { inserted. { l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I've put in what seems to be necessary to fix the current column of the current alignment. Try to go on, since this might almost work. ! Missing } inserted. } l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing $ inserted. $ l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Missing } inserted. } l.409 ...dth=.3\columnwidth]{2MM/FIG/square2.pdf}} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Misplaced alignment tab character &. l.410 & \multicolumn{1}{m{.5\columnwidth}}{\centering\includegraphics[width=.... I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Misplaced \omit. \multispan ->\omit \@multispan l.410 ...dth=.3\columnwidth]{2MM/FIG/square5.pdf}} I expect to see \omit only after tab marks or the \cr of an alignment. Proceed, and I'll ignore this case. ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.410 ...dth=.3\columnwidth]{2MM/FIG/square5.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. ...nextchar \ifcase \@chclass \@classz \or \@classi \or \@classii... l.410 ...dth=.3\columnwidth]{2MM/FIG/square5.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Illegal character in array arg. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.410 ...dth=.3\columnwidth]{2MM/FIG/square5.pdf}} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. ...nextchar \ifcase \@chclass \@classz \or \@classi \or \@classii... l.410 ...dth=.3\columnwidth]{2MM/FIG/square5.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! Undefined control sequence. \@mkpream ...hclass }\ifcase \@lastchclass \@acol \or \or \@preamerr \@ne \o... l.410 ...dth=.3\columnwidth]{2MM/FIG/square5.pdf}} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: There's no line here to end. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.411 \\ Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.411 \\ I've inserted a begin-math/end-math symbol since I think you left one out. Proceed, with fingers crossed. Overfull \hbox (20.0pt too wide) in paragraph at lines 404--411 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.412 ( a) & (b) You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced alignment tab character &. l.412 (a) & (b) I can't figure out why you would want to use a tab mark here. If you just want an ampersand, the remedy is simple: Just type `I\&' now. But if some right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. Missing character: There is no ( in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ) in font nullfont! ! Misplaced \crcr. \endtabular ->\crcr \egroup \egroup $\egroup l.413 \end{tabular} $ I can't figure out why you would want to use a tab mark or \cr or \span just now. If something like a right brace up above has ended a previous alignment prematurely, you're probably due for more error messages, and you might try typing `S' now just to see what is salvageable. ! Extra }, or forgotten \endgroup. \endtabular ->\crcr \egroup \egroup $\egroup l.413 \end{tabular} $ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Extra }, or forgotten \endgroup. \endtabular ->\crcr \egroup \egroup $\egroup l.413 \end{tabular} $ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! Extra }, or forgotten $. \egroup l.413 \end{tabular} $ I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! LaTeX Error: \begin{document} ended by \end{tabular}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.413 \end{tabular} $ Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.413 \end{tabular} $ I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! LaTeX Error: \caption outside float. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.415 \caption {(a) Illustration of Lemma~\ref{clm1}. (b) Crossing edges $(... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. ! Undefined control sequence. ...ing: Reference `clm1' on page \thepage \space undefined\on@line . l.415 ...tion{(a) Illustration of Lemma~\ref{clm1} . (b) Crossing edges $(a,b... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `clm1' on page undefined on input line 415. ! Missing } inserted. } l.415 ...of Lemma~\ref{clm1}. (b) Crossing edges $ (a,b)$ and $(c,d)$ form an... I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. Missing character: There is no ( in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no ) in font nullfont! Missing character: There is no ( in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no ) in font nullfont! ! Extra }, or forgotten $. l.415 ...dges $(a,b)$ and $(c,d)$ form an anchor.} I've deleted a group-closing symbol because it seems to be spurious, as in `$x}$'. But perhaps the } is legitimate and you forgot something else, as in `\hbox{$x}'. In such cases the way to recover is to insert both the forgotten and the deleted material, e.g., by typing `I$}'. ! LaTeX Error: \begin{document} ended by \end{figure}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.418 \end{figure} Your command was ignored. Type I to replace it with another command, or to continue without it. ! Missing $ inserted. $ l.418 \end{figure} I've inserted something that you may have forgotten. (See the above.) With luck, this will get me unwedged. But if you really didn't forget anything, try typing `2' now; then my insertion and my current dilemma will both disappear. ! Extra \endgroup. \endgroup l.418 \end{figure} Things are pretty mixed up, but I think the worst is over. Overfull \hbox (288.63536pt too wide) in paragraph at lines 412--419 []$$\OML/cmm/m/it/10 and$$formananchor:[]$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.420 I n the proof of Lemma~\ref{clm1}, $a$ is connected to $b$, $c$, and $d... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...ing: Reference `clm1' on page \thepage \space undefined\on@line . l.420 In the proof of Lemma~\ref{clm1} , $a$ is connected to $b$, $c$, and $d... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `clm1' on page undefined on input line 420. Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! 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Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 420--421 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 420--421 [] [] Overfull \hbox (5.28589pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 a$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 b$ [] Overfull \hbox (4.32756pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 c$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 d$ [] Overfull \hbox (5.28589pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 a$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 b$ [] Overfull \hbox (4.32756pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 c$ [] Overfull \hbox (5.20486pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 d$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 420--421 \OML/cmm/m/it/10 r$ [] ! LaTeX Error: Environment corollary undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.422 \begin{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.424 T he endpoints of every two crossing edges in $\dg$ are at distance at ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no M in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{corollary}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.425 \end{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 424--426 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 424--426 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (9.78938pt too wide) in paragraph at lines 424--426 \/cmr/m/n/10 2\OML/cmm/m/it/10 r$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 424--426 \OML/cmm/m/it/10 r$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.427 B ased on the proof of Lemma~\ref{clm1}, we define an {\em anchor} as a... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no B in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...ing: Reference `clm1' on page \thepage \space undefined\on@line . l.427 Based on the proof of Lemma~\ref{clm1} , we define an {\em anchor} as a... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `clm1' on page undefined on input line 427. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! 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Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no F in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! ! Undefined control sequence. ...ference `fig:square1' on page \thepage \space undefined\on@line . l.428 $cd$; see Figure~\ref{fig:square1} (b). We call $a$ as the {\em crown} ... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. 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Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! 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Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! ! Undefined control sequence. ...ing: Reference `clm1' on page \thepage \space undefined\on@line . l.429 In view of Lemma~\ref{clm1} , two crossing The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `clm1' on page undefined on input line 429. Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 427--431 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 427--431 [] [] Overfull \hbox (14.31102pt too wide) in paragraph at lines 427--431 \OT1/cmr/m/it/10 an- [] Overfull \hbox (19.03876pt too wide) in paragraph at lines 427--431 \OT1/cmr/m/it/10 chor [] Overfull \hbox (42.44324pt too wide) in paragraph at lines 427--431 \OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 a; b; c; d\OMS/cmsy/m/n/10 g$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (23.70721pt too wide) in paragraph at lines 427--431 \OMS/cmsy/m/n/10 4\OML/cmm/m/it/10 cad$ [] Overfull \hbox (4.29166pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 b$ [] Overfull \hbox (5.28589pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 a$ [] Overfull \hbox (9.53242pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 cd$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 427--431 [] [] Overfull \hbox (5.28589pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 a$ [] Overfull \hbox (25.68315pt too wide) in paragraph at lines 427--431 \OT1/cmr/m/it/10 crown [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 r$ [] Overfull \hbox (8.61922pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 bc$ [] Overfull \hbox (9.49652pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 bd$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 427--431 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 427--431 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] ! LaTeX Error: Environment observation undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.432 \begin{observation} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.434 $ \dg$ is plane if and only if it has no anchor. You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{observation}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.435 \end{observation} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (50.87642pt too wide) in paragraph at lines 434--436 []$\OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] ! LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.437 \begin{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.439 $ n^{-\frac{2}{3}}$ is a threshold for $\dg$ to be plane. You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.440 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.441 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ...eference `anchor-obs' on page \thepage \space undefined\on@line . l.442 ...is plane, by Observation~\ref{anchor-obs} , it is enough to show tha... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `anchor-obs' on page undefined on input line 442. Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no E in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...nce `connected-k-thr' on page \thepage \space undefined\on@line . l.442 ...nnected. By Theorem~\ref{connected-k-thr} , if $r=o(n^{-\frac{2}{3}}... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `connected-k-thr' on page undefined on input line 442 . Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! 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Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (38.55519pt too wide) in paragraph at lines 439--444 []$\OML/cmm/m/it/10 n[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 439--444 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 439--444 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 439--444 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 439--444 [] [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 439--444 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (31.18022pt too wide) in paragraph at lines 439--444 \OML/cmm/m/it/10 o\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 439--444 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 439--444 \/cmr/m/n/10 4$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.445 W e prove the second statement by adjusting the proof of Theorem~\ref{c... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no j in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...nce `connected-k-thr' on page \thepage \space undefined\on@line . l.445 ...he proof of Theorem~\ref{connected-k-thr} for $k=4$. Assume The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `connected-k-thr' on page undefined on input line 445 . Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no L in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! 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Overfull \hbox (19.99419pt too wide) in paragraph at lines 452--453 $\OML/cmm/m/it/10 c[] > [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 452--453 \/cmr/m/n/10 0$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 452--453 [] [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 452--453 \OML/cmm/m/it/10 X$ [] Overfull \hbox (196.65787pt too wide) detected at line 453 []\/cmr/m/n/10 (\OML/cmm/m/it/10 X\/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ^^T \OML/cmm/ m/it/10 c[]n[]r[] \/cmr/m/n/10 + \OML/cmm/m/it/10 c[]n[]r[] \/cmr/m/n/10 + \OML /cmm/m/it/10 c[]n[]r[] \/cmr/m/n/10 + \OML/cmm/m/it/10 c[]n[]r[]: [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.455 S ince $r=\omega(n^{-\frac{2}{3}})$, $\frac{\Var (X)}{\Ex[X]^2}$ tends ... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no 0 in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no O in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! ! Undefined control sequence. ...eference `anchor-obs' on page \thepage \space undefined\on@line . l.456 By Observation~\ref{anchor-obs} , w.h.p. $\dg$ is not plane. The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `anchor-obs' on page undefined on input line 456. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.457 \qed\end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 455--458 [] [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 455--458 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (32.9163pt too wide) in paragraph at lines 455--458 \OML/cmm/m/it/10 !\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (28.12025pt too wide) in paragraph at lines 455--458 []$ [] Overfull \hbox (6.00235pt too wide) in paragraph at lines 455--458 \OML/cmm/m/it/10 n$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 455--458 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 455--458 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 455--458 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.69054pt too wide) in paragraph at lines 455--458 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.460 A s a direct consequence of the proof of Theorem~\ref{plane-thr}, we ha... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...Reference `plane-thr' on page \thepage \space undefined\on@line . l.460 ...e of the proof of Theorem~\ref{plane-thr} , we have the following: The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `plane-thr' on page undefined on input line 460. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no : in font nullfont! ! LaTeX Error: Environment corollary undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.461 \begin{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no W in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{corollary}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.463 \end{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 460--464 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 460--464 [] [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.466 N ote that every anchor introduces a crossing and each crossing introdu... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no S in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no y in font nullfont! ! Undefined control sequence. l.466 ...cted graph and has four points, by~\eqref {E-X} we have the followin... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. Missing character: There is no E in font nullfont! Missing character: There is no - in font nullfont! Missing character: There is no X in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: Environment corollary undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.467 \begin{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{corollary}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.470 \end{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 466--471 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 466--471 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (35.31958pt too wide) in paragraph at lines 466--471 \/cmr/m/n/10 ^^B(\OML/cmm/m/it/10 n[]r[]\/cmr/m/n/10 )$ [] ! Undefined control sequence. l.472 \subsection {The threshold for $\dg$ to be planar} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.472 \subsection{T he threshold for $\dg$ to be planar} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 472--474 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 472--474 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.475 I n this section we investigate the threshold for the planarity of a ra... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no I in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no K in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no k in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no ' in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no ; in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...ing: Reference `thm3' on page \thepage \space undefined\on@line . l.475 ...ent as in the proof of Theorem~\ref{thm3} we can show that if $r\ge... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `thm3' on page undefined on input line 475. Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 475--476 [] [] Overfull \hbox (12.97919pt too wide) in paragraph at lines 475--476 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (19.34032pt too wide) in paragraph at lines 475--476 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 475--476 [] [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 475--476 \OML/cmm/m/it/10 r \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (42.9989pt too wide) in paragraph at lines 475--476 []$ [] Overfull \hbox (4.78937pt too wide) in paragraph at lines 475--476 \OML/cmm/m/it/10 r$ [] Overfull \hbox (12.97919pt too wide) in paragraph at lines 475--476 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 475--476 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] ! LaTeX Error: Environment theorem undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.477 \begin{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.479 $ n^{-\frac{5}{8}}$ is a threshold for $\dg$ to be planar. You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{theorem}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.480 \end{theorem} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Environment proof undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.481 \begin{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no B in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...erence `clique-k-thr' on page \thepage \space undefined\on@line . l.482 By Theorem~\ref{clique-k-thr} , if $r=\omega(n^{-\frac{5}{8}})$, then w... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `clique-k-thr' on page undefined on input line 482. Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! 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LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.487 N ote that, in order to prove that $\dg$ is planar, we have to show tha... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no N in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! 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Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{proof}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.489 \qed\end{proof} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 487--490 [] [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (12.97919pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (19.34032pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (12.97919pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (19.34032pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 487--490 \/cmr/m/n/10 5$ [] Overfull \hbox (35.37737pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 n[] < [] Overfull \hbox (46.16223pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 n[]$ [] Overfull \hbox (16.07635pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 k \OMS/cmsy/m/n/10 ^^U [] Overfull \hbox (5.00002pt too wide) in paragraph at lines 487--490 \/cmr/m/n/10 5$ [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 487--490 [] [] Overfull \hbox (15.34488pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 r \/cmr/m/n/10 = [] Overfull \hbox (31.18022pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 o\/cmr/m/n/10 (\OML/cmm/m/it/10 n[]\/cmr/m/n/10 )$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (12.97919pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (19.34032pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 K[]$ [] Overfull \hbox (30.87642pt too wide) in paragraph at lines 487--490 \OML/cmm/m/it/10 G\/cmr/m/n/10 (\OML/cmm/m/it/10 n; r\/cmr/m/n/10 )$ [] Overfull \hbox (5.69054pt too wide) in paragraph at lines 487--490 [] [] ! LaTeX Error: The font size command \normalsize is not defined: there is probably something wrong with the class file. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.490 Your command was ignored. Type I to replace it with another command, or to continue without it. [2 ! Undefined control sequence. \newlabel{thm:clique-2mm}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{thm:clique-2mm}{{}{}} ! Undefined control sequence. \newlabel{eqn:clique-var}{{1}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{eqn:clique-var}{{1}{}} ! Undefined control sequence. \newlabel{eqn:clique-var-2}{{2}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{eqn:clique-var-2}{{2}{}} ! Undefined control sequence. ...abel{eqn:clique-var-ineq}{{3}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{eqn:clique-var-ineq}{{3}{}} ! Undefined control sequence. ...label{eqn:clique-var-sum}{{4}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{eqn:clique-var-sum}{{4}{}} ! Undefined control sequence. \newlabel{eqn:edgesScupT}{{5}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{eqn:edgesScupT}{{5}{}} ! Undefined control sequence. ...abel{eqn:clique-var-sum2}{{6}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{eqn:clique-var-sum2}{{6}{}} ! Undefined control sequence. \newlabel{eqn:ineq1}{{7}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{eqn:ineq1}{{7}{}} ! Undefined control sequence. \newlabel{connected-section}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{connected-section}{{}{}} ! Undefined control sequence. \newlabel{connected-k-thr}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{connected-k-thr}{{}{}} ! Undefined control sequence. \newlabel{X-Xi}{{8}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{X-Xi}{{8}{}} ! Undefined control sequence. \newlabel{E-X-Xi}{{9}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{E-X-Xi}{{9}{}} ! Undefined control sequence. \newlabel{Xi-upper}{{10}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{Xi-upper}{{10}{}} ! Undefined control sequence. \newlabel{Xi-lower}{{11}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{Xi-lower}{{11}{}} ! Undefined control sequence. \newlabel{E-Xi}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{E-Xi}{{}{}} ! Undefined control sequence. \newlabel{E-X}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{E-X}{{}{}} ! Undefined control sequence. \newlabel{1mom2:eq}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{1mom2:eq}{{}{}} ! Undefined control sequence. \newlabel{sS-fig}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{sS-fig}{{}{}} ! Undefined control sequence. \newlabel{Chebyshev}{{12}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{Chebyshev}{{12}{}} ! Undefined control sequence. \newlabel{Var-Ex2}{{13}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{Var-Ex2}{{13}{}} ! Undefined control sequence. \newlabel{Var-X}{{14}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{Var-X}{{14}{}} ! Undefined control sequence. \newlabel{Var-X-2}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{Var-X-2}{{}{}} ! Undefined control sequence. \newlabel{E-XiXj}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{E-XiXj}{{}{}} ! Undefined control sequence. \newlabel{E-XiXj-2}{{15}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{E-XiXj-2}{{15}{}} ! Undefined control sequence. \newlabel{Var-Ex2-2}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{Var-Ex2-2}{{}{}} ! Undefined control sequence. \newlabel{clique-k-thr}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{clique-k-thr}{{}{}} ! Undefined control sequence. \newlabel{plane-section}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{plane-section}{{}{}} ! Undefined control sequence. \newlabel{fig:square}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{fig:square}{{}{}} ! Undefined control sequence. \newlabel{thm3}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{thm3}{{}{}} ! Undefined control sequence. \newlabel{clm1}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{clm1}{{}{}} ! Undefined control sequence. \newlabel{fig:square1}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{fig:square1}{{}{}} ! Undefined control sequence. \newlabel{2-hop-cor}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{2-hop-cor}{{}{}} ! Undefined control sequence. \newlabel{anchor-obs}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{anchor-obs}{{}{}} ! Undefined control sequence. \newlabel{plane-thr}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{plane-thr}{{}{}} ! Undefined control sequence. \newlabel{cr:cor}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{cr:cor}{{}{}} ! Undefined control sequence. \newlabel{planar-section}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{planar-section}{{}{}} ! Undefined control sequence. \newlabel{planarity-thr}{{}{\thepage }} l.490 The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. \newlabel{planarity-thr}{{}{}} ] ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.491 A s a direct consequence of the proof of Theorem~\ref{planarity-thr}, w... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no A in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...rence `planarity-thr' on page \thepage \space undefined\on@line . l.491 ... the proof of Theorem~\ref{planarity-thr} , we have the following: The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `planarity-thr' on page undefined on input line 491. Missing character: There is no , in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no : in font nullfont! ! LaTeX Error: Environment corollary undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.492 \begin{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Missing character: There is no W in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no q in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{corollary}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.494 \end{corollary} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 491--495 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 491--495 [] [] ! Undefined control sequence. l.497 \subsection {Conclusions} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.497 \subsection{C onclusions} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no C in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no W in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no z in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Missing character: There is no A in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no T in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no m in font nullfont! ! Undefined control sequence. ...nce `connected-k-thr' on page \thepage \space undefined\on@line . l.498 ...s to extend Theorem~\ref{connected-k-thr} for connected subgraphs o... The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. LaTeX Warning: Reference `connected-k-thr' on page undefined on input line 498 . Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no y in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no u in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no g in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no . in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 497--499 [] [] Overfull \hbox (10.86105pt too wide) in paragraph at lines 497--499 [] [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 497--499 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 497--499 \OML/cmm/m/it/10 k$ [] Overfull \hbox (5.52084pt too wide) in paragraph at lines 497--499 \OML/cmm/m/it/10 k$ [] Overfull \hbox (4.8229pt too wide) in paragraph at lines 497--499 \OML/cmm/m/it/10 ^^N$ [] ! Undefined control sequence. l.500 \section {Exercises} The control sequence at the end of the top line of your error message was never \def'ed. If you have misspelled it (e.g., `\hobx'), type `I' and the correct spelling (e.g., `I\hbox'). Otherwise just continue, and I'll forget about whatever was undefined. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.500 \section{E xercises} You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no E in font nullfont! Missing character: There is no x in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no c in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Overfull \hbox (20.0pt too wide) in paragraph at lines 500--501 [] [] ! LaTeX Error: Environment exercise undefined. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.502 \begin{exercise} Your command was ignored. Type I to replace it with another command, or to continue without it. ! LaTeX Error: Missing \begin{document}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.503 S how that for two independent random variables, $X$ and $Y$, $Var[X+Y]... You're in trouble here. Try typing to proceed. If that doesn't work, type X to quit. Missing character: There is no S in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no h in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no f in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no w in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no p in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no t in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no o in font nullfont! Missing character: There is no m in font nullfont! Missing character: There is no v in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no r in font nullfont! Missing character: There is no i in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no b in font nullfont! Missing character: There is no l in font nullfont! Missing character: There is no e in font nullfont! Missing character: There is no s in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no a in font nullfont! Missing character: There is no n in font nullfont! Missing character: There is no d in font nullfont! Missing character: There is no , in font nullfont! Missing character: There is no . in font nullfont! ! LaTeX Error: \begin{document} ended by \end{exercise}. See the LaTeX manual or LaTeX Companion for explanation. Type H for immediate help. ... l.504 \end{exercise} Your command was ignored. Type I to replace it with another command, or to continue without it. Overfull \hbox (20.0pt too wide) in paragraph at lines 503--505 [] [] Overfull \hbox (9.06943pt too wide) in paragraph at lines 503--505 \OML/cmm/m/it/10 X$ [] Overfull \hbox (8.02779pt too wide) in paragraph at lines 503--505 \OML/cmm/m/it/10 Y$ [] Overfull \hbox (37.75583pt too wide) in paragraph at lines 503--505 \OML/cmm/m/it/10 Var\/cmr/m/n/10 [\OML/cmm/m/it/10 X \/cmr/m/n/10 + [] Overfull \hbox (21.36108pt too wide) in paragraph at lines 503--505 \OML/cmm/m/it/10 Y\/cmr/m/n/10 ] = [] Overfull \hbox (40.53362pt too wide) in paragraph at lines 503--505 \OML/cmm/m/it/10 Var\/cmr/m/n/10 [\OML/cmm/m/it/10 X\/cmr/m/n/10 ] + [] Overfull \hbox (31.71417pt too wide) in paragraph at lines 503--505 \OML/cmm/m/it/10 Var\/cmr/m/n/10 [\OML/cmm/m/it/10 Y\/cmr/m/n/10 ]$ [] ) ! Emergency stop. <*> UDG-Thresholds-Fixed.tex *** (job aborted, no legal \end found) Here is how much of TeX's memory you used: 96 strings out of 493117 1237 string characters out of 6135433 110512 words of memory out of 5000000 3654 multiletter control sequences out of 15000+600000 4294 words of font info for 16 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 18i,8n,25p,846b,144s stack positions out of 5000i,500n,10000p,200000b,80000s ! ==> Fatal error occurred, no output PDF file produced!