Distinguishing subgroups of the rationals by their Ramsey properties

TL;DR

This paper explores the Ramsey properties of subgroups of the rationals, showing an infinite hierarchy of subgroups distinguished by their partition regular systems, extending previous results about the differences between Z, Q, and R.

Contribution

It demonstrates an uncountable chain of subgroups of Q with distinct partition regular properties, generalizing earlier findings about Z and Q.

Findings

Existence of an infinite chain of subgroups of Q with unique partition regular systems

Partition regularity distinguishes subgroups of Q based on their ring containment

The chain of subgroups can be uncountably infinite

Abstract

A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some time that there is an infinite system of linear equations that is partition regular over R but not over Q, and it was recently shown (answering a long-standing open question) that one can also distinguish Q from Z in this way. Our aim is to show that the transition from Z to Q is not sharp: there is an infinite chain of subgroups of Q, each of which has a system that is partition regular over it but not over its predecessors. We actually prove something stronger: our main result is that if R and S are subrings of Q with R not contained in S, then there is a system that is partition regular over R but not over S. This implies, for example, that the chain above may be taken to be uncountable.