\documentclass[12pt,oneside,openany]{book} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \DeclareFontFamily{U}{rsfs}{\skewchar\font127 } \DeclareFontShape{U}{rsfs}{m}{n}{<-6.5>rsfs5<6.5-7.5>rsfs7<7.5->rsfs10}{} \usepackage{booktabs,array} \usepackage[numbers,sort&compress]{natbib} \usepackage{environ} \PassOptionsToPackage{hypertexnames=false}{hyperref} \input{style} \usepackage{xurl}% \usepackage[nameinlink,capitalize,noabbrev]{cleveref} \usepackage{aliascnt} \hypersetup{pdftitle={Existence of the q-Colorability Threshold for Sparse Random Graphs},pdfauthor={P. M. Aronow and Patrick Lopatto}} \renewcommand{\chaptername}{Section} \numberwithin{equation}{chapter} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[chapter] \newaliascnt{proposition}{theorem} \newtheorem{proposition}[proposition]{Proposition} \aliascntresetthe{proposition} \newaliascnt{lemma}{theorem} \newtheorem{lemma}[lemma]{Lemma} \aliascntresetthe{lemma} \newaliascnt{corollary}{theorem} \newtheorem{corollary}[corollary]{Corollary} \aliascntresetthe{corollary} \theoremstyle{definition} \newaliascnt{definition}{theorem} \newtheorem{definition}[definition]{Definition} \aliascntresetthe{definition} \newaliascnt{example}{theorem} \newtheorem{example}[example]{Example} \aliascntresetthe{example} \theoremstyle{remark} \newaliascnt{remark}{theorem} \newtheorem{remark}[remark]{Remark} \aliascntresetthe{remark} \makeatletter \@addtoreset{proposition}{chapter}\@addtoreset{lemma}{chapter}\@addtoreset{corollary}{chapter} \@addtoreset{definition}{chapter}\@addtoreset{example}{chapter}\@addtoreset{remark}{chapter} \makeatother \crefname{theorem}{Theorem}{Theorems} \crefname{proposition}{Proposition}{Propositions} \crefname{lemma}{Lemma}{Lemmas} \crefname{corollary}{Corollary}{Corollaries} \crefname{definition}{Definition}{Definitions} \crefname{example}{Example}{Examples} \crefname{remark}{Remark}{Remarks} \crefname{chapter}{Section}{Sections} \crefname{appendix}{Appendix}{Appendices} \crefname{part}{Part}{Parts} \crefname{section}{Section}{Sections} \crefname{subsection}{Section}{Sections} \crefname{table}{Table}{Tables} \crefname{equation}{}{} \crefformat{equation}{(#2#1#3)} \Crefformat{equation}{(#2#1#3)} \crefrangeformat{equation}{(#3#1#4)--(#5#2#6)} \Crefrangeformat{equation}{(#3#1#4)--(#5#2#6)} \crefmultiformat{equation}{(#2#1#3)}{ and~(#2#1#3)}{, (#2#1#3)}{, and~(#2#1#3)} \Crefmultiformat{equation}{(#2#1#3)}{ and~(#2#1#3)}{, (#2#1#3)}{, and~(#2#1#3)} \newcommand{\creflastconjunction}{, and~} \newcommand{\creflastgroupconjunction}{, and~} \makeatletter \newcommand{\manuscript@noteblock}[2]{% \ifx\relax#1\relax\else{\centering\manuscript@spacedsc{#1}\par}\vspace{6pt}\fi {\centering \begin{minipage}{0.84\textwidth}\small\normalfont\setlength{\parindent}{1.5em}% \noindent\ignorespaces#2\end{minipage}\par}} \NewEnviron{authorsnote}{% \clearpage\thispagestyle{empty}% \vspace*{\stretch{2}}% \manuscript@noteblock{Authors' note}{\BODY}% \vspace*{\stretch{3}}% \clearpage} \NewEnviron{abstractpage}{% \clearpage\thispagestyle{empty}% \vspace*{\stretch{2}}% \manuscript@noteblock{Abstract}{\BODY}% \vspace*{\stretch{3}}% \clearpage} \newcommand{\manuscripttitlepage}[2]{% \hypersetup{pageanchor=false}% \begin{titlepage} \centering\normalfont \vspace*{1.7in} {\large\strut\par} \vspace{1.05in} {\fontsize{28}{35}\selectfont #1\par} \vspace{1.1in} {\Large\def\\{\par\vspace{10pt}}#2\par} \vfill \end{titlepage}% \hypersetup{pageanchor=true}} \makeatother \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,shapes.geometric,calc,fit,backgrounds,decorations.pathreplacing} \input{macros} \allowdisplaybreaks[2] \setcounter{tocdepth}{0} \clubpenalty=10000 \patchcmd{\thebibliography}{\clubpenalty4000}{\clubpenalty10000}{}{} \begin{document} \frontmatter \manuscripttitlepage{Existence of the\\ $q$-Colorability Threshold\\ for Sparse Random Graphs}{P.\ M.\ Aronow\\ Patrick Lopatto} \begin{authorsnote} With the exception of these notes, the entire document is machine-written. We have carefully supervised its preparation to ensure that the proofs are readable and that previous literature is appropriately cited. We are releasing this manuscript on Hexagon to disseminate the results as quickly as possible because they may be of interest to the mathematical community. A more carefully written version of this manuscript is in preparation and will be submitted for publication. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas. P.\ L.\ was partially supported by NSF grant DMS-2450004. \end{authorsnote} \begin{abstractpage} \input{abstract} \end{abstractpage} \tableofcontents \mainmatter \input{sections/01-intro} \input{sections/02-overview} \input{sections/body} \backmatter \setlength{\bibsep}{3pt plus 1pt minus 1pt} \bibliographystyle{plainnat} \bibliography{refs} \end{document}