Abstract colorings, games and ultrafilters

TL;DR

This paper unifies various Ramsey-type theorems related to colorings of complete graphs on infinite semigroups using ultrafilters and topological games, connecting multiple fields and recent breakthroughs.

Contribution

It provides a general framework that encompasses several known theorems across different areas using ultrafilters and topological game techniques.

Findings

Unified generalization of Ramsey-type theorems

Application of ultrafilters in semigroup colorings

Connection to covering properties and topological games

Abstract

The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for natural numbers due to Milliken--Tylor, Deuber--Hindman, Bergelson--Hindman, for combinatorial covering properties due to Scheepers and Tsaban, and local properties in function spaces due to Scheepers. To this end, we use idempotent ultrafilters in the \v{C}ech--Stone compactifications of discrete infinite semigroups and topological games. The research is motivated by the recent breakthrough work of Tsaban about colorings and the Menger covering property.