Van der Waerden type theorem for amenable groups and FC-groups

TL;DR

This paper extends Van der Waerden type theorems to certain classes of groups, showing that monochromatic configurations exist in finitely colored amenable and FC-groups, partially confirming a conjecture by Bergelson and McCutcheon.

Contribution

It proves a Van der Waerden type theorem for discrete, countable, amenable groups and FC-groups, advancing understanding of combinatorial properties in these algebraic structures.

Findings

In finitely colored amenable groups, specific monochromatic configurations are left IP*

The result extends to FC-groups for higher powers of the group

Partially confirms a conjecture of Bergelson and McCutcheon

Abstract

We prove that for a discrete, countable, and amenable group $G$, if the direct product $G^2=G \times G$ is finitely colored then $\{ g \in G : \text{exists } (x,y) \in G^2 \text{ such that } \{ (x,y),(xg,y),(xg,yg)\} \text{ is monochromatic} \}$, is left IP$^{\ast}$. This partially solves a conjecture of V. Bergelson and R. McCutcheon. Moreover, we prove that the result holds for $G^m$ if $G$ is an FC-group, i.e., all conjugacy classes of $G$ are finite.