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

TL;DR

This paper establishes that for sufficiently large n (at least 10^11), any 2-coloring of the n-dimensional grid [4]^n necessarily contains a monochromatic combinatorial line, providing a new upper bound for the Hales-Jewett number HJ(4,2).

Contribution

The paper proves a concrete upper bound for HJ(4,2) for large n, improving understanding of the Hales-Jewett theorem in high dimensions.

Findings

Monochromatic line exists for n ≥ 10^11 in [4]^n under 2-coloring.

Provides an explicit large bound for HJ(4,2).

Advances the quantitative understanding of the Hales-Jewett number.

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.