Almost $\mathcal{R}$-trivial monoids are almost never Ramsey

TL;DR

This paper investigates the conditions under which naturally occurring monoids, especially almost r-trivial monoids, are Ramsey, concluding that most are not, except in trivial cases, using combinatorial analysis of their structure.

Contribution

It introduces a combinatorial criterion to determine when almost r-trivial monoids are Ramsey, highlighting that most naturally occurring monoids do not satisfy this property.

Findings

Most geometric monoids are not Ramsey.

A simple combinatorial condition can check Ramsey property.

Almost r-trivial monoids are rarely Ramsey in natural cases.

Abstract

Recent results have generalized Gowers' Theorem (to Lupini's Theorem) and the Furstenberg-Katznelson theorem, both infinite dimensional Ramsey Theorems. The framework of arXiv:1611.06600 provides a machine which accepts (almost $\mathcal{R}$-trivial) monoids and outputs Ramsey theorems. The major generalization was to extract monoid actions from these theorems. We investigate the other direction, and feed into the machine monoids which appear "naturally", and which are not extracted from a Ramsey theorem, such as such as $0$-Hecke monoids and hyperplane face monoids. Most examples of monoids coming from geometry will not be Ramsey monoids. We provide a simple combinatorial condition for checking this that goes through the representation of almost $\mathcal{R}$-trivial monoids as a family of (almost) regressive functions. Except in extremely small or trivial cases, the naturally occurring monoids will not be Ramsey.