Ultrafilters: Where topological dynamics = algebra = combinatorics

TL;DR

This paper explores the deep connections between topological dynamics, ultrafilter semigroups, and combinatorics, demonstrating their interplay through a proof of the Hales-Jewett theorem.

Contribution

It provides a survey of the relationships among these fields and offers a new proof of the Hales-Jewett theorem using ultrafilter techniques.

Findings

Ultrafilter methods unify topological dynamics and combinatorics.

A new proof of the Hales-Jewett theorem is presented.

Connections between algebraic and topological structures are elucidated.

Abstract

We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem.