## 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.

Original language | English |
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Pages (from-to) | 93-104 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 129 |

DOIs | |

Publication status | Published - 15 Oct 2014 |

## Keywords

- Central sets
- Partion regular
- Rationals
- Subgroups