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.