Partition regularity in the rationals

TL;DR

This paper introduces a new infinite partition regular system of linear equations over the rationals, featuring unbounded coefficients, and distinguishes it from previous systems by its unique properties and solutions.

Contribution

It presents the first example of a partition regular system with a variable having unbounded coefficients and resolves an open problem about systems that are partition regular over Q but not over N.

Findings

First example of a partition regular system with unbounded coefficients

System is partition regular over Q but not over N

Introduces a genuinely different type of partition regular system

Abstract

A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of an infinite partition regular system of equations. Since then, other such systems of equations have been found, but each can be viewed as a modification of the Finite Sums theorem. We present here a new infinite partition regular system of equations that appears to arise in a genuinely different way. This is the first example of a partition regular system in which a variable occurs with unbounded coefficients. A modification of the system provides an example of a system that is partition regular over Q but not N, settling another open problem.