## Abstract

A system of homogeneous linear equations with integer coefficients is

We present here a new infinite partition regular system of equations that appears to arise in a genuinely different way. This is the first example of a partition regular system in which a variable occurs with unbounded coefficients. A modification of the system provides an example of a system that is partition regular over

*partition regular*if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of an infinite partition regular system of equations. Since then, other such systems of equations have been found, but each can be viewed as a modification of the Finite Sums theorem.We present here a new infinite partition regular system of equations that appears to arise in a genuinely different way. This is the first example of a partition regular system in which a variable occurs with unbounded coefficients. A modification of the system provides an example of a system that is partition regular over

**Q**but not**N**, settling another open problem.Original language | English |
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Pages (from-to) | 1590–1599 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 120 |

Issue number | 7 |

Early online date | 12 Jun 2013 |

DOIs | |

Publication status | Published - 1 Sep 2013 |

## Keywords

- Partition regularity
- Ramsey theory
- Finite Sums theorem