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

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.