Enumeration of monochromatic three term arithmetic progressions in two-colorings of any finite group

TL;DR

This paper develops a novel approach using real algebraic geometry and semidefinite programming to count monochromatic three-term arithmetic progressions in any finite group, extending beyond abelian groups.

Contribution

It introduces a new computational method to enumerate monochromatic arithmetic progressions in finite groups, applicable to both abelian and non-abelian groups.

Findings

Derived lower bounds for the number of monochromatic progressions

Applied advanced computational techniques to non-abelian groups

Extended enumeration results beyond traditional abelian cases

Abstract

There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to obtain. In this research project we do not only determine existence, but study the more general problem of counting them. We formulate the enumeration problem as a problem in real algebraic geometry and then use state of the art computational methods in semidefinite programming and representation theory to derive lower bounds for the number of monochromatic arithmetic progressions in any finite group.