Density Hales-Jewett and Moser numbers

For any $n\ge 0$ and $k\ge 1$, the \emph{density Hales-Jewett number} $c_{n,k}$ is defined as the size of the largest subset of the cube $[k{]}^n$ := $\{1,\mathrm{.}\mathrm{.}\mathrm{.},k{\}}^n$ which contains no combinatorial line; similarly, the Moser number $c_{n,k}^{\mathrm{\prime }}$ 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 $n\to \mathrm{\infty }$ (which implies a similar claim for $c_{n,k}^{\mathrm{\prime }}$); 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 $c_{n,k}^{\mathrm{\prime }}$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,\mathrm{.}\mathrm{.}\mathrm{.},6$ is $1,2,6,18,52,150,450$, while the sequence $c_{n,3}^{\mathrm{\prime }}$ for $n=0,\mathrm{.}\mathrm{.}\mathrm{.},6$ 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$\ell$isthelargestintegersuchthat$2k > 2^\ell$.