Concerning partition regular matrices

TL;DR

This paper explores the properties of infinite image partition regular matrices, especially their behavior under diagonal sums, and introduces new classes of such matrices to generate more examples.

Contribution

It extends the theory of image partition regular matrices by analyzing their diagonal sums involving Milliken-Taylor and centrally image partition regular matrices.

Findings

Diagonal sum of Milliken-Taylor and certain centrally image partition regular matrices is also image partition regular.

Centrally image partition regular matrices are closed under diagonal sum.

New classes of image partition regular matrices are constructed.

Abstract

A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb N$ is finitely colored, there must be some $\vec{x}$ with entries from $\mathbb N$ such that all entries of $A\vec{x}$ are in some color class. In [6] and [8], the notion of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite image partition regular matrices. It was shown that centrally image partition regular matrices are closed under diagonal sum. In the present paper, we show that the diagonal sum of two matrices, one of which comes from the class of all Milliken-Taylor matrices and the other from a suitable subclass of the class of all centrally image partition regular matrices is also image partition regular. This will produce more image partition regular matrices. We also study the multiple structures within one cell of a finite partition of $\mathbb N$ $\cdot$