Density Hales-Jewett and Moser numbers

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 \(2k > 2^\ell\).