The Wildcard Set's Size in the Density Hales-Jewett Theorem

Abstract

In this paper prove results concerning restrictions on the cardinality of the wildcard set in the density Hales-Jewett theorem, establishing in particular that for general $k$ one may choose this cardinality from any IP set and that for $k=2$ it may be chosen to be a square, thus providing an abstract extension of Sárközy's theorem on square differences in sets of positive upper density.