Ramsey-product subsets of a group

We say that a subset $S$ of an infinite group $G$ is a Ramsey-product subset if, for any infinite subsets $X$, $Y$ of $G$, there exist $x\in X$ and $y\in Y$suchthat$x y \in S$and$ y x \in S$\mathrm{.}Weshowthatthefamily$\varphi$ofallRamsey-productsubsetsof$G$isafilterand$\varphi$ defines the subsemigroup $\overline{G^{\ast }{G}^{\ast }}$ of the semigroup $G^{\ast }$ of all free ultrafilters on $G$.