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Matrix Formulation of Moreira Theorem

Sayan Goswami

TL;DR

The paper establishes a matrix analogue of Moreira's Ramsey-type result by showing that for finite image partition regular matrices $A$ and $B$ of the same order, every finite coloring of $\mathbb{N}$ admits vectors $\overrightarrow{X},\overrightarrow{Y}$ such that $\{A\overrightarrow{X},\\ A\overrightarrow{X}+B\overrightarrow{Y},\\ A\overrightarrow{X}\cdot B\overrightarrow{Y}\}$ is monochromatic, with coordinate-wise addition and multiplication. A stronger formulation asserts the monochromaticity of $\{a,a+b,ab : a\in A\overrightarrow{X}, b\in B\overrightarrow{Y}\}$ as a whole. The proof blends a compactness argument with ultrafilter techniques in the Stone-Čech compactification, exploiting minimal idempotents and central sets to secure a common monochromatic image and then lift it to the stated triple. The work extends Ramseyan partition regularity to a matrix setting and recovers the original Moreira result when $A=B=(1)$, while noting that the infinite-matrix case remains open.

Abstract

In a celebrated article, Moreira proved for every finite coloring of the set of naturals, there exists a monochromatic copy of the form $\{x,x+y,xy\},$ which gives a partial answer to one of the central open problems of Ramsey theory asking whether $\{x,y,x+y,xy\}$ is partition regular. In this article, we prove the matrix version of the Moreira theorem. We prove that if $A$ and $B$ are two finite image partition regular matrices of the same order, then for every finite coloring of the set of naturals, there exist two vectors $\overrightarrow{X}, \overrightarrow{Y}$ such that $\{A\overrightarrow{X}, A\overrightarrow{X}+B\overrightarrow{Y}, A \overrightarrow{X}\cdot B\overrightarrow{Y}\}$ is monochromatic, where addition and multiplication are defined coordinate-wise.

Matrix Formulation of Moreira Theorem

TL;DR

The paper establishes a matrix analogue of Moreira's Ramsey-type result by showing that for finite image partition regular matrices and of the same order, every finite coloring of admits vectors such that is monochromatic, with coordinate-wise addition and multiplication. A stronger formulation asserts the monochromaticity of as a whole. The proof blends a compactness argument with ultrafilter techniques in the Stone-Čech compactification, exploiting minimal idempotents and central sets to secure a common monochromatic image and then lift it to the stated triple. The work extends Ramseyan partition regularity to a matrix setting and recovers the original Moreira result when , while noting that the infinite-matrix case remains open.

Abstract

In a celebrated article, Moreira proved for every finite coloring of the set of naturals, there exists a monochromatic copy of the form which gives a partial answer to one of the central open problems of Ramsey theory asking whether is partition regular. In this article, we prove the matrix version of the Moreira theorem. We prove that if and are two finite image partition regular matrices of the same order, then for every finite coloring of the set of naturals, there exist two vectors such that is monochromatic, where addition and multiplication are defined coordinate-wise.

Paper Structure

This paper contains 2 sections, 3 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['todo']}

Key Result

Theorem 1.2

For any $r\in \mathbb{N},$$F\in \mathcal{P}_f(\mathbb{P})$ and for any $r$- coloring $\mathbb{N}=\bigcup_{i=1}^rC_i$ there exists $i\in [1,r]$ and $y\in \mathbb{N}$ such that is infinite. In particular $\{x,xy,x+y\}$ is partition regular.

Theorems & Definitions (5)

  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['todo']} assuming Lemma \ref{['lem']}
  • proof : Proof of Lemma \ref{['lem']}