Matrix Formulation of Moreira Theorem

TL;DR

This paper extends Moreira's theorem to a matrix setting, showing that for two finite image partition regular matrices, certain structured sets are monochromatic under any finite coloring of natural numbers.

Contribution

It introduces a matrix formulation of Moreira's theorem, establishing conditions for monochromatic sets involving matrix transformations and coordinate-wise operations.

Findings

Proves the matrix version of Moreira's theorem.

Shows existence of monochromatic sets involving matrix images and operations.

Extends partition regularity results to matrix frameworks.

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.