\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\citation{Sier}
\citation{R}
\citation{FFK}
\citation{C1}
\citation{C2}
\citation{C3}
\citation{C4}
\citation{FFK}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{Intro}{{1}{1}{Introduction}{section.1}{}}
\citation{Bang}
\citation{Z}
\citation{BV}
\citation{BHV}
\@writefile{toc}{\contentsline {section}{\numberline {2}Definitions and Preliminaries}{2}{section.2}}
\newlabel{Bang}{{2.2}{2}{}{thm.2.2}{}}
\citation{Guy}
\citation{E}
\citation{Sier}
\newlabel{Erd}{{2}{3}{Definitions and Preliminaries}{Item.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Sierpi\'{n}ski's Theorem}{4}{section.3}}
\newlabel{Sierpinski}{{3.1}{4}{}{thm.3.1}{}}
\citation{Guy}
\citation{I}
\citation{C3}
\citation{C4}
\citation{FFK}
\citation{C4}
\citation{FFK}
\citation{I}
\@writefile{toc}{\contentsline {section}{\numberline {4}Variations of Sierpi\'{n}ski's Theorem}{5}{section.4}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Some Recent Variations}{5}{subsection.4.1}}
\newlabel{Chen}{{4.1}{5}{}{thm.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Some New Variations}{5}{subsection.4.2}}
\newlabel{Jsection}{{4.2}{5}{Some New Variations}{subsection.4.2}{}}
\newlabel{J1}{{4.2}{6}{}{thm.4.2}{}}
\newlabel{J1A}{{4.4}{6}{}{thm.4.4}{}}
\citation{S}
\citation{ST}
\newlabel{Conjecture}{{4.6}{7}{}{thm.4.6}{}}
\newlabel{C1}{{4.7}{7}{}{thm.4.7}{}}
\newlabel{kernel}{{4.8}{7}{}{thm.4.8}{}}
\newlabel{Sch1}{{4.9}{7}{}{thm.4.9}{}}
\newlabel{C2}{{4.10}{7}{}{thm.4.10}{}}
\newlabel{C3}{{4.11}{8}{}{thm.4.11}{}}
\newlabel{Jmaxshort1}{{4.12}{8}{}{thm.4.12}{}}
\newlabel{Jmax}{{4.12}{8}{}{thm.4.12}{}}
\newlabel{easycase1}{{1}{8}{}{Item.3}{}}
\newlabel{composite}{{2}{8}{}{Item.4}{}}
\newlabel{easycase2andSier}{{3}{8}{}{Item.5}{}}
\newlabel{misoddprime}{{4}{8}{}{Item.6}{}}
\newlabel{Schinz}{{5}{8}{}{Item.7}{}}
\newlabel{specialq}{{4.13}{9}{}{thm.4.13}{}}
\newlabel{Eq1}{{1}{9}{Some New Variations}{equation.4.1}{}}
\newlabel{rem2}{{4.2}{13}{}{equation.4.1}{}}
\bibcite{Bang}{1}
\bibcite{BHV}{2}
\bibcite{BV}{3}
\bibcite{C1}{4}
\bibcite{C2}{5}
\bibcite{C3}{6}
\bibcite{C4}{7}
\@writefile{toc}{\contentsline {section}{\numberline {5}Acknowledgments}{14}{section.5}}
\bibcite{E}{8}
\bibcite{FFK}{9}
\bibcite{Guy}{10}
\bibcite{I}{11}
\bibcite{R}{12}
\bibcite{S}{13}
\bibcite{ST}{14}
\bibcite{Sier}{15}
\bibcite{Z}{16}