Another Note on Intervals in the Hales–Jewett Theorem

Abstract

The Hales–Jewett Theorem states that any $r$–colouring of $[m]^n$ contains a monochromatic combinatorial line if $n$ is large enough. Shelah's proof of the theorem implies that for $m = 3$ there always exists a monochromatic combinatorial line whose set of active coordinates is the union of at most $r$ intervals. For odd $r$, Conlon and Kamčev constructed $r$–colourings for which it cannot be fewer than $r$ intervals. However, we show that for even $r$ and large $n$, any $r$–colouring of $[3]^n$ contains a monochromatic combinatorial line whose set of active coordinates is the union of at most $r-1$ intervals. This is optimal and extends a result of Leader and Räty for $r=2$.