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

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


\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}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}



\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.