A Note on Intervals in the Hales-Jewett Theorem

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.