Polymath.tex

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

\documentclass[12pt,a4paper,reqno]{amsart} \usepackage{amssymb} \usepackage{amscd} %\usepackage{psfig} %\usepackage{showkeys} % uncomment this when editing cross-references \numberwithin{equation}{section}

    \addtolength{\textwidth}{3 truecm}
\setlength{\voffset}{-.6 truecm}
\setlength{\hoffset}{-1.3 truecm}



\theoremstyle{plain}

\newtheorem{theorem}{Theorem}[section] %\newtheorem{theorem}[theorem]{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{principle}[theorem]{Principle} \newtheorem{claim}[theorem]{Claim}

\theoremstyle{definition}

%\newtheorem{roughdef}[subsection]{Rough Definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{example}[theorem]{Example} \newtheorem{examples}[theorem]{Examples} %\newtheorem{problem}[subsection]{Problem} %\newtheorem{question}[subsection]{Question}

\newcommand\E{\mathbb{E}} \renewcommand\P{\mathbb{P}}

\parindent 0mm \parskip 5mm

\begin{document}

\title{Density Hales-Jewett and Moser numbers in low dimensions}

\subjclass{???}

\begin{abstract} For any $n \geq 0$ and $k \geq 1$, the density Hales-Jewett number $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n$ := $\{1,\ldots,k\}^n$ which contains no combinatorial line; similarly, the Moser number $c'_{n,k}$ is the largest subset of the cube $[k]^n$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson \cite{fk1}, \cite{fk2}, \cite{mcc} shows that $c_{n,k}$ = $o(k^n)$ as $n \to \infty$ (which implies a similar claim for $c'_{n,k}$; this is already non-trivial for $k = 3$. Several new proofs of this result have also been recently established \cite{poly}, \cite{austin}.

Using both human and computer-assisted arguments, we compute several values of $c_{n,k}$ and $c'n,k$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,\ldots,6$ is $1,2,6,18,52,150,450$, while the sequence $c'_{n,3}$ for $n=0,\ldots,5$ is $1,2,6,16,43,124$. We also establish some results for higher $k$, showing for instance that an analogue of the LYM inequality (which relates to the $k = 2$ case) does not hold for higher $k$. \end{abstract}

\maketitle %\today

\setcounter{tocdepth}{1} \tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction}

...

In Sections \ref{moser-lower-sec}, \ref{moser-upper-sec} we will show

\begin{theorem}[Values of $c'_{n,3}$ for small $n$]\label{moser} We have $c'_{0,3} = 1$, $c'_{1,3} = 2$, $c'_{2,3} = 6$, $c'_{3,3} = 16$, $c'_{4,3} = 43$, $c'_{5,3} = 124$, and $353 \leq c'_{6,3} \leq 361$. \end{theorem}

\begin{remark} The values of $c'_{n,3}$ for $n=0,1,2$ are easily verified. The quantity $c'_{3,3}$ was first computed in \cite{chvatal2}, while the quantity $c'_{4,3}$ was computed in \cite{chandra}; we will give alternate proofs of these results here. The bounds for $c'_{5,3}$ and $c'_{6,3}$ are new. \end{remark}

\section{Lower bounds for the Moser problem}\label{moser-lower-sec}

...

(Here we prove the lower bounds in Theorem \ref{moser}.)

\section{Upper bounds for the $k=3$ Moser problem in small dimensions}\label{moser-upper-sec}

\include{moser}

\begin{thebibliography}{10}

\bibitem{austin} T. Austin, \emph{Deducing the density Hales-Jewett theorem from an infinitary removal lemma}, preprint.

\bibitem{chandra} A. Chandra, \emph{On the solution of Moser's problem in four dimensions}, Canad. Math. Bull. \textbf{16} (1973), 507--511.

\bibitem{chvatal1} V. Chvatal, \emph{Remarks on a problem of Moser}, Canadian Math Bulletin, Vol 15, 1972, 19--21.

\bibitem{chvatal2} V. Chvatal, \emph{Edmonds polytopes and a hierarchy of combinatorial problems}, Discrete Math. 4 (1973) 305-337.

\bibitem{fk1} H. Furstenberg, Y. Katznelson, \emph{A density version of the Hales-Jewett theorem for k = 3}, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–-241.

\bibitem{fk2} H. Furstenberg, Y. Katznelson, \emph{A density version of the Hales-Jewett theorem}, J. Anal. Math. 57 (1991), 64–-119. MR1191743

\bibitem{komlos} J. Komlos, solution to problem P.170 by Leo Moser, Canad. Math.. Bull. vol (??check) (1972), 312-313, 1970.

\bibitem{moser} L. Moser, Problem P.170 in Canad. Math. Bull. 13 (1970), 268.

\bibitem{mcc} R. McCutcheon, \emph{The conclusion of the proof of the density Hales-Jewett theorem for k=3}, unpublished.

\bibitem{poly} D.H.J. Polymath, ???, preprint.

\end{thebibliography}

\end{document}