Intervals in the Hales-Jewett theorem

Abstract

The Hales-Jewett theorem states that for any $m$ and $r$ there exists an $n$ such that any $r$-colouring of the elements of $[m]^n$ contains a monochromatic combinatorial line. We study the structure of the wildcard set $S \subseteq [n]$ which determines this monochromatic line, showing that when $r$ is odd there are $r$-colourings of $[3]^n$ where the wildcard set of a monochromatic line cannot be the union of fewer than $r$ intervals. This is tight, as for $n$ sufficiently large there are always monochromatic lines whose wildcard set is the union of at most $r$ intervals.