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Ramsey's witnesses

Mauro Di Nasso, Lorenzo Luperi Baglini, Marcello Mamino, Rosario Mennuni, Mariaclara Ragosta

TL;DR

The paper generalises partition regularity to infinitary Ramsey configurations using ultrafilter and nonstandard methods, introducing Ramsey's witnesses (RW) and Ramsey pairs to capture when infinite monochromatic structures must arise under finite colourings. A central nonstandard characterisation shows that Ramsey PR is equivalent to the existence of RW with prescribed properties, enabling powerful dichotomies between sums, products, exponentials, and polynomial equations. The work demonstrates that many classical PR results fail under the Ramsey lens (e.g., no Ramsey version of van der Waerden's theorem; two-variable polynomials are not Ramsey PR), while some forms reduce to Schur-type equations (e.g., $x+y=z$ and $-x+y=z$) under typical RW constraints. The authors also map a landscape of block-partition regularity for equations like $\varphi(x,y,z,t)$, guided by Moreira-type results, and articulate open problems including a Puritz-like theory for Ramsey pairs and extensions to broader polynomial forms.

Abstract

We introduce the notion of Ramsey partition regularity, a generalisation of partition regularity involving infinitary configurations. We provide characterisations of this notion in terms of certain ultrafilters related to tensor products and dubbed Ramsey's witnesses; and we also consider their nonstandard counterparts as pairs of hypernatural numbers, called Ramsey pairs. These characterisations are then used to determine whether various configurations involving polynomials and exponentials are Ramsey partition regular over the natural numbers.

Ramsey's witnesses

TL;DR

The paper generalises partition regularity to infinitary Ramsey configurations using ultrafilter and nonstandard methods, introducing Ramsey's witnesses (RW) and Ramsey pairs to capture when infinite monochromatic structures must arise under finite colourings. A central nonstandard characterisation shows that Ramsey PR is equivalent to the existence of RW with prescribed properties, enabling powerful dichotomies between sums, products, exponentials, and polynomial equations. The work demonstrates that many classical PR results fail under the Ramsey lens (e.g., no Ramsey version of van der Waerden's theorem; two-variable polynomials are not Ramsey PR), while some forms reduce to Schur-type equations (e.g., and ) under typical RW constraints. The authors also map a landscape of block-partition regularity for equations like , guided by Moreira-type results, and articulate open problems including a Puritz-like theory for Ramsey pairs and extensions to broader polynomial forms.

Abstract

We introduce the notion of Ramsey partition regularity, a generalisation of partition regularity involving infinitary configurations. We provide characterisations of this notion in terms of certain ultrafilters related to tensor products and dubbed Ramsey's witnesses; and we also consider their nonstandard counterparts as pairs of hypernatural numbers, called Ramsey pairs. These characterisations are then used to determine whether various configurations involving polynomials and exponentials are Ramsey partition regular over the natural numbers.

Paper Structure

This paper contains 8 sections, 37 theorems, 24 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Ramsey's witnesses
  3. Ramsey partition regularity
  4. Pairwise sums and products: a case study
  5. Ramsey PR of equations
  6. Case $v(\alpha)=v(\beta)$.
  7. Case $v(\alpha)\ll v(\beta)$.
  8. Open problems

Key Result

Theorem 1

For every finite colouring of $\mathbb{N}$ there exists an infinite increasing sequence $(x_i)_{i\in \mathbb N}$ such that the set of all finite sums $\text{FS}((x_i)_{i\in \mathbb N})=\{x_{i_1}+\ldots+x_{i_k}\mid k\in \mathbb N, i_1<\ldots<i_k\}$ is monochromatic.

Theorems & Definitions (102)

  • Theorem : Hindman's Theorem
  • Theorem : Ramsey's Theorem
  • Theorem : Pairwise Sum-Product Theorem
  • Definition 1.2
  • Theorem A
  • Theorem B: \ref{['cor: poly applications', 'pr:ramseydiff']}
  • Theorem C: \ref{['no infinite 3-AP']}
  • Definition 2.1
  • Remark 2.4
  • Remark 2.5
  • ...and 92 more