Ben Barber, , Neil Hindman, Imre Leader and Dona Strauss, Journal of Combinatorial Theory, Series A, Volume 129, January 2015, Pages 93–104 PDF
In Partition regularity in the rationals we (Barber, Hindman and Leader) showed that there are systems of equations that are partition regular over 
but not over 
. Here we show that this separation is very strong: there is an uncountable chain of subgroups from to such that each subgroup has a system that is partition regular over it but over no earlier subgroup. We use our new central sets approach, but this result could also have been proved using the original density method.
Most of the work in this paper is spent proving that the systems we construct are strongly partition regular, in the sense that the variables can be forced to take different values. If you only want to see the application then you can skip part of the argument without losing anything.