An upper bound for the Hales-Jewett number HJ(4,2)

Abstract

We show that for $n$ at least $10^{11}$, any 2-coloring of the $n$-dimensional grid $[4]^n$ contains a monochromatic combinatorial line. This is a special case of the Hales-Jewett Theorem, to which the best known general upper bound is due to Shelah; Shelah's recursion gives an upper bound between $2 \uparrow \uparrow 7$ and $2 \uparrow \uparrow 8$ for the case we consider, and no better value was previously known.