Matrix Formulation of Moreira Theorem

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.