Asymptotic Ramsey theory of Diophantine equations

2025/10/22 by Lorenzo Luperi Baglini, Baglini, Lorenzo Luperi, Alessandro Vegnuti +1
Computer Science · Mathematics · #03H15 #05A17 #11U10 #54D80 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical and Theoretical Analysis #Primary: 05D10. Secondary: 26E35

paper · pdf · doi:10.48550/arxiv.2510.19370

openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

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.

Citations

• The magic of tensor products of ultrafilters
• Ramsey's witnesses
• Diophantine equations with three monomials
• Rado functionals and applications
• A limiting result for the Ramsey theory of functional equations
• Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
• Ramsey properties of nonlinear Diophantine equations
• Fermat-like equations that are not partition regular
• Hypernatural Numbers as Ultrafilters
• Iterated hyper-extensions and an idempotent ultrafilter proof of Rado's\n theorem
• Partition Regularity of Nonlinear Polynomials: a Nonstandard Approach

Related

• Ramsey properties of nonlinear Diophantine equations
• A limiting result for the Ramsey theory of functional equations
• Ramsey's witnesses
• Hypernatural numbers in arithmetic Ramsey theory
• On the Effectiveness of Partition Regularity over Algebraic Structures