# Polymath.tex

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\title{Density Hales-Jewett and Moser numbers}

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\begin{abstract} For any $n \geq 0$ and $k \geq 1$, the \emph{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 $n=0,\ldots,6$ is $1,2,6,16,43,124,353$. We also prove 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 $k$, and also establishing the asymptotic lower bound $c_{n,k} \geq k^n \exp\left( - O(\sqrt[\ell]{\log n})\right)$ where $\ell$ is the largest integer such that $2k > 2^\ell$. \end{abstract}

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\begin{thebibliography}{10}

\bibitem{ajtai} M. Ajtai, E. Szemer\'edi, \emph{Sets of lattice points that form no squares}, Studia Scientiarum Mathematicarum Hungarica, \textbf{9} (1974-1975), 9--11.

\bibitem{austin} T. Austin, \emph{Deducing the density Hales-Jewett theorem from an infinitary removal lemma}, preprint, available at {\tt arxiv.org/abs/0903.1633}.

\bibitem{beck} J. Beck, Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press, 2008, Cambridge.

\bibitem{behrend} F. Behrend, \emph{On the sets of integers which contain no three in arithmetic progression}, Proceedings of the National Academy of Sciences \textbf{23} (1946), 331–-332.

\bibitem{Brower} A. Brower, {\tt www.win.tue.nl/$\sim$aeb/codes/binary-1.html}.

\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. Chv\'{a}tal, \emph{Remarks on a problem of Moser}, Canad. Math. Bull., \textbf{15} (1972) 19--21.

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

\bibitem{elkin} M. Elkin, \emph{An Improved Construction of Progression-Free Sets}, preprint.

\bibitem{fuji} K. Fujimura, {\tt www.puzzles.com/PuzzlePlayground/CoinsAndTriangles/CoinsAndTriangles.htm}

\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. \textbf{75} (1989), 227–-241.

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

\bibitem{kra} D. Geller, I. Kra, S. Popescu, S. Simanca, \emph{On circulant matrices}, {\tt www.math.sunysb.edu/$\sim$sorin/eprints/circulant.pdf}

\bibitem{greenwolf} B. Green, J. Wolf, \emph{A note on Elkin's improvement of Behrend's construction}, preprint, available at {\tt arxiv.org/abs/0810.0732}.

\bibitem{heule} M. Heule, presentation at {\tt www.st.ewi.tudelft.nl/sat/slides/waerden.pdf}

\bibitem{komlos} J. Koml\'{o}s, solution to problem P.170 by Leo Moser, Canad. Math. Bull. \textbf{15} (1972), 312--313, 1970.

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

\bibitem{markstrom} K. Markstrom, Template:\tt abel.math.umu.se/$\sim$klasm/Data/HJ/

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

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

\bibitem{obryant} K. O'Bryant, \emph{Sets of integers that do not contain long arithmetic progressions}, preprint, available at {\tt arxiv.org/abs/0811.3057}.

\bibitem{oeis} N. J. A. Sloane, Ed. (2008), The On-Line Encyclopedia of Integer Sequences, {\tt www.research.att.com/$\sim$njas/sequences/}

\bibitem{potenchin} A. Potechin, \emph{Maximal caps in $AG(6, 3)$}, Des. Codes Cryptogr., \textbf{46} (2008), 243--259.

\bibitem{poly} D.H.J. Polymath, \emph{A new proof of the density Hales-Jewett theorem}, preprint, available at {\tt arxiv.org/abs/0910.3926}.

\bibitem{polywiki} D.H.J. Polymath, {\tt michaelnielsen.org/polymath1/index.php?title=Polymath1}

\bibitem{rankin} R. A. Rankin, \emph{Sets of integers containing not more than a given number of terms in arithmetical progression}, Proc. Roy. Soc. Edinburgh Sect. A \textbf{65} (1960/1961), 332–-344.

\bibitem{roth} K. Roth, \emph{On certain sets of integers, I}, J. Lond. Math. Soc. \textbf{28} (1953), 104-–109.

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

\bibitem{shelah} S. Shelah, \emph{Primitive recursive bounds for van der Warden numbers}, J. Amer. Math. Soc. \textbf{28} (1988), 683-–697.

\bibitem{sperner} E. Sperner, \emph{Ein Satz \"uber Untermengen einer endlichen Menge}, Mathematische Zeitschrift \textbf{27} (1928), 544-–548.

\bibitem{szem} E. Szemer\'edi, \emph{On sets of integers containing no $k$ elements in arithmetic progression}, Acta Arithmetica \textbf{27} (1975), 199-–245.

\end{thebibliography}

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