Monoid actions and ultrafilter methods in Ramsey theory

TL;DR

This paper develops algebraic and dynamical frameworks involving monoid actions and ultrafilters to derive new and stronger results in Ramsey theory, including a novel proof of the Furstenberg--Katznelson theorem.

Contribution

It introduces algebraic structures for infinitary Ramsey statements and connects them with monoid actions to prove generalized Ramsey theorems.

Findings

Proves a theorem on monoid actions by endomorphisms of semigroups.

Introduces algebraic structures formalizing infinitary Ramsey statements.

Provides a new, stronger proof of the Furstenberg--Katznelson Ramsey theorem.

Abstract

First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg--Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers' Ramsey theorem.