On Hales-Jewett and Related Numbers

We investigate the Hales-Jewett numbers and some variants of them, both computationally and via enumerative and probabilistic arguments.

In particular, we give the improved lower bound formula on the combinatorial-geometric variant of the Hales-Jewett numbers. {As a consequence of one of the variants investigated, we show that the Milton Bradley game Connect Four played with 2 players on a 5-dimensional hypercube of any side length can end in a draw.