Infinite image partition regular matrices - Solution in C-sets

TL;DR

This paper introduces C-image partition regular matrices, a new subclass of infinite matrices that extend finite image partition regularity results, and demonstrates their relation to known classes.

Contribution

It defines C-image partition regular matrices, expanding the theory of infinite image partition regular matrices and connecting them with existing classes.

Findings

Many known centrally image partition regular matrices are C-image partition regular.

C-image partition regular matrices form an interesting subclass with specific properties.

The paper extends finite image partition regularity concepts to infinite matrices.

Abstract

A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb{N}$ is finitely colored, there must be some $\overset{\rightarrow}{x}$ with entries from $\mathbb{N}$ such that all entries of $A \overset{\rightarrow}{x}$ are in the same color class. Comparing to the finite case, infinite image partition regular matrices seem more harder to analyze. The concept of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite one. In this paper, we shall introduce the notion of C-image partition regular matrices, an interesting subclass of centrally image partition regular matrices. Also we shall see that many of known centrally image partition regular matrices are C-image partition regular.