# Ramsey-product subsets of a group

## TL;DR

This paper introduces the concept of Ramsey-product subsets in infinite groups, showing they form a filter and relate to a specific subsemigroup of ultrafilters, advancing the understanding of algebraic and combinatorial properties of groups.

## Contribution

It defines Ramsey-product subsets in groups, proves they form a filter, and links them to a subsemigroup of ultrafilters, providing new insights into group combinatorics.

## Key findings

- The family of all Ramsey-product subsets forms a filter.
- Ramsey-product subsets define a subsemigroup of ultrafilters.
- The structure of these subsets relates to algebraic properties of the 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 $x y \in S$ and $ y x \in S$ . We show that the family $\varphi$ of all Ramsey-product subsets of $G$ is a filter and $\varphi$ defines the subsemigroup $ \overline{G^*G^*}$ of the semigroup $G^*$ of all free ultrafilters on $G$.