Distinguishing subgroups of the rationals by their Ramsey properties

A system of linear equations with integer coefficients is partition regular over a subset S   of the reals if, whenever S∖{0}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 RR but not over QQ, and it was recently shown (answering a long-standing open question) that one can also distinguish QQ from ZZ in this way.Our aim is to show that the transition from ZZ to QQ is not sharp: there is an infinite chain of subgroups of QQ, 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 QQ 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.