  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.
