Monochromatic non-commuting products

TL;DR

This paper proves that in any finite coloring of an amenable group, there are many monochromatic sets with non-commuting products, extending non-commutative Schur theorems using quasirandom colorings.

Contribution

It introduces quasirandom colorings and provides the first combinatorial proof of non-commutative Schur theorems for amenable groups.

Findings

Existence of many monochromatic non-commuting product sets in amenable groups

Introduction of quasirandom colorings as a key tool

Extension of non-commutative Schur theorem to broader group classes

Abstract

We show that a finite coloring of an amenable group contains `many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables. This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem. Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.