Density Hales-Jewett and Moser numbers

TL;DR

This paper investigates the density Hales-Jewett and Moser numbers, providing new computational results for small parameters, analyzing their asymptotic behavior, and exploring properties for higher values of k.

Contribution

It offers the first computed values for small n and k, proves that certain inequalities do not extend to higher k, and establishes new asymptotic lower bounds for the numbers.

Findings

Computed specific values of c_{n,k} and c'_{n,k} for small n and k.

Showed that the LYM inequality does not hold for higher k.

Established a new asymptotic lower bound for c_{n,k}.

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 $[k]^n$ := $\{1,...,k\}^n$ 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 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. 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 $n=0,...,6$ is $1,2,6,18,52,150,450$, while the sequence $c'_{n,3}$ for $n=0,...,6$ 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 $c_{n,k} \geq k^n \exp(- O(\sqrt[\ell]{\log n}))$ where $\ell$ is the largest integer such that $2k > 2^\ell$.