Van der Waerden type theorem for amenable groups and FC-groups (2411.15987v1)

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.