A Note on Intervals in the Hales-Jewett Theorem

TL;DR

This paper proves that for the Hales-Jewett cube with alphabet size 3, any 2-coloring necessarily contains a monochromatic line with an interval as its active coordinate set, disproving a previous conjecture.

Contribution

It demonstrates that the conjecture claiming the existence of 2-colorings without interval-active monochromatic lines is false.

Findings

Any 2-coloring of [3]^n contains a monochromatic line with an interval active set.

Disproves Conlon and Kamcev's conjecture about the absence of such lines.

Supports the idea that interval-active monochromatic lines are unavoidable in 2-colorings.

Abstract

The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube [3]^n is r-coloured there is a monochromatic line (for n large). Conlon and Kamcev conjectured that, for any n, there is a 2-colouring of [3]^n for which there is no monochromatic line whose active coordinate set is an interval. In this note we disprove this conjecture.