Asymptotic Ramsey theory of Diophantine equations

TL;DR

The paper introduces asymptotic partition regularity (PR) for Diophantine equations and connects it to Archimedean classes of hypernaturals via nonstandard analysis and $u$-equivalence, providing a unifying framework for Ramsey-type results. It proves that PR is equivalent to asymptotic PR in a structured sense and develops tools to translate polynomial equations into Archimedean-class configurations, enabling streamlined proofs of classical results (Rado-type theorems) and new negative results for nonlinear equations. A central achievement is a strong, general necessary condition for Fermat–Catalan-type equations, constraining when nonlinear PR can hold. The work further links these ideas to ultrafilters on $\beta\mathbb{N}$, introducing Archimedean closeness of ultrafilters and showing how asymptotic reasoning yields concise proofs of known ultrafilter equations, with several open questions guiding future research on the number of asymptotic classes and their ultrafilter analogues.

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.

Theorems & Definitions (123)

• Definition 1
• Theorem : A
• Theorem : B
• Theorem : C
• Theorem 1.1
• proof
• Example 1.2
• Theorem 1.3: HindmanLeaderIneq, Theorem 2
• Theorem 1.4
• proof