Intervals in the Hales-Jewett theorem

TL;DR

This paper investigates the structure of wildcard sets in the Hales-Jewett theorem, revealing tight bounds on the number of intervals needed to form monochromatic lines in certain colorings.

Contribution

It characterizes the wildcard sets in monochromatic lines for specific colorings, showing tight bounds on the number of intervals needed, especially when the number of colors is odd.

Findings

For odd r, there exist r-colorings of [3]^n where the wildcard set cannot be the union of fewer than r intervals.

For large n, monochromatic lines always have wildcard sets that are the union of at most r intervals.

The results establish tight bounds on the interval structure of wildcard sets 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.