Ramsey theory for layered semigroups

TL;DR

This paper develops a unified framework using layered semigroups and nonstandard analysis to prove various classical Ramsey theorems, simplifying proofs and extending results in combinatorial and topological Ramsey theory.

Contribution

It introduces a general framework for proving Ramsey statements via layered semigroups and nonstandard methods, unifying and extending several key theorems in the field.

Findings

Unified proof of Gowers' $ ext{FIN}_k$ theorem

Nonstandard proof of Graham-Rothschild theorem

Generalization of Hindman's finite sums theorem

Abstract

We further develop the theory of layered semigroups, as introduced by Farah, Hindman and McLeod, providing a general framework to prove Ramsey statements about such a semigroup $S$. By nonstandard and topological arguments, we show Ramsey statements on $S$ are implied by the existence of "coherent" sequences in $S$. This framework allows us to formalise and prove many results in Ramsey theory, including Gowers' $\mathrm{FIN}_k$ theorem, the Graham-Rothschild theorem, and Hindman's finite sums theorem. Other highlights include: a simple nonstandard proof of the Graham-Rothschild theorem for strong variable words; a nonstandard proof of Bergelson-Blass-Hindman's partition theorem for located variable words, using a result of Carlson, Hindman and Strauss; and a common generalisation of the latter result and Gowers' theorem, which can be proven in our framework.