Asymptotic Ramsey theory of Diophantine equations

TL;DR

This paper develops a new framework for understanding the asymptotic behavior of Diophantine equations in Ramsey theory, linking partition regularity with nonstandard analysis to derive both known and novel results.

Contribution

It introduces the concept of asymptotic partition regularity for Diophantine equations and applies nonstandard analysis to produce new results, including for Fermat-Catalan equations.

Findings

Asymptotic partition regularity is central to negative results in Ramsey theory of equations.

The framework connects partition regularity with nonstandard analysis.

New results are obtained for Fermat-Catalan equations.

Abstract

We introduce the notion of asymptotic partition regularity for Diophantine equations. We show how this notion is at the core of almost all known negative results in the Ramsey theory of equations, and we use it to produce new ones, as in the case of Fermat-Catalan equations. The methods we use here are based on translating asymptotic partition regularity into the context of nonstandard extensions, via the notion of Archimedean equivalence classes of hypernaturals.