Ramsey-product subsets of a group

Abstract

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 $y\in Y$ such that $xy\in S$ and $yx\in S$ . We show that the family $\phi$ of all Ramsey-product subsets of $G$ is a filter and $\phi$ defines the subsemigroup $\overline{G^{\ast }{G}^{\ast }}$ of the semigroup $G^{\ast }$ of all free ultrafilters on $G$.