Inhomogeneous Partition Regularity

TL;DR

This paper proves that the characterization of inhomogeneous partition regularity of linear systems over integers extends to general rings, confirming that such systems are partition regular if and only if they have a constant solution.

Contribution

It provides a new direct proof of Rado's theorem and extends the characterization of partition regularity to arbitrary rings.

Findings

Partition regularity over rings matches the integer case.

A new direct proof of Rado's theorem is established.

The result confirms the equivalence of partition regularity and constant solutions over rings.

Abstract

We say that the system of equations $Ax=b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax=b$. Rado proved that the system $Ax=b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new `direct' proof of Rado's result.