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arXiv · 1405.0460

Distinguishing subgroups of the rationals by their Ramsey properties

Ben Barber·Neil Hindman·Imre Leader·Dona Strauss
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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.

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Ben Barber, Neil Hindman, Imre Leader, Dona Strauss. 2014-05-02. Distinguishing subgroups of the rationals by their Ramsey properties. https://doi.org/10.1016/j.jcta.2014.10.002

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