# Difference between revisions of "Polymath.tex"

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\section{Introduction} | \section{Introduction} | ||

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− | + | \section{Lower bounds for the density Hales-Jewett problem}\label{dhj-lower-sec} | |

− | \ | + | \include{dhj-lown-lower} |

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− | \ | + | \section{Upper bounds for the $k=3$ density Hales-Jewett problem}\label{dhj-upper-sec} |

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+ | \include{dhj-lown} | ||

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

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

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− | + | \section{Higher values $k$} | |

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+ | \include{higherk} | ||

\appendix | \appendix | ||

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\include{genetic} | \include{genetic} | ||

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+ | \section{Integer programming} | ||

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\begin{thebibliography}{10} | \begin{thebibliography}{10} |

## Revision as of 11:37, 27 May 2009

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

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\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}

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\begin{document}

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

\author{D.H.J. Polymath} \address{http://michaelnielsen.org/polymath1/index.php} \email{???}

\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}

\include{introduction}

\section{Lower bounds for the density Hales-Jewett problem}\label{dhj-lower-sec}

\include{dhj-lown-lower}

\section{Upper bounds for the $k=3$ density Hales-Jewett problem}\label{dhj-upper-sec}

\include{dhj-lown}

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

\include{moser-lower}

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

\include{moser}

\section{Fujimura's problem}

\include{fujimura}

\section{Higher values $k$}

\include{higherk}

\appendix

\section{Genetic algorithms}

\include{genetic}

\section{Integer programming}

\include{integer}

\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{Krisha} K. Krishna, M. Narasimha Murty, "Genetic K-means algorithm," Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on , vol.29, no.3, pp.433-439, Jun 1999

\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.

\bibitem{Rothlauf} F. Rothlauf, D. E. Goldberg, Representations for Genetic and Evolutionary Algorithms. Physica-Verlag, 2002.

\end{thebibliography}

\end{document}