## Abstract

A system of linear equations with integer coefficients is

Our aim is to show that the transition from

*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 |

Early online date | 15 Oct 2014 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

## Keywords

- Partion regular
- Rationals
- Subgroups
- Central sets