An Upper Bound for the Hales--Jewett Number $\mathrm{HJ}(4,2)$

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 [Hales and Jewett, Trans. Amer. Math., 106 (1963), pp. 222--229], to which the best known general upper bound is due to Shelah [J. Amer. Math. Soc., 1 (1988), pp. 683--697]; 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.