  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 $2k >
2^\ell$.
