Enumeration of monochromatic three term arithmetic progressions in two-colorings of cyclic groups

TL;DR

This paper develops a computational approach using semidefinite programming and algebraic geometry to count monochromatic three-term arithmetic progressions in two-colorings of cyclic groups, providing explicit bounds.

Contribution

It introduces a novel method combining real algebraic geometry and computational techniques to enumerate and bound monochromatic progressions in cyclic groups.

Findings

Derived explicit lower bounds for monochromatic progressions

Reformulated enumeration problem in algebraic geometry framework

Applied advanced computational methods to cyclic groups of any order

Abstract

One of the toughest problems in Ramsey theory is to determine the existence of monochromatic arithmetic progressions in groups whose elements have been colored. We study the harder problem to not only determine the existence of monochromatic arithmetic progressions, but to also count them. We reformulate the enumeration in real algebraic geometry and then use state of the art computational methods in semidefinite programming and representation theory to derive sharp, or an explicit constant from sharp, lower bounds for the cyclic group of any order.