Partition regularity in the rationals

Ben Barber, Neil Hindman, Imre Leader

Research output: Contribution to journalArticle (Academic Journal)peer-review

5 Citations (Scopus)
251 Downloads (Pure)


A system of homogeneous linear equations with integer coefficients is 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 languageEnglish
Pages (from-to)1590–1599
Number of pages10
JournalJournal of Combinatorial Theory, Series A
Issue number7
Early online date12 Jun 2013
Publication statusPublished - 1 Sep 2013


  • Partition regularity
  • Ramsey theory
  • Finite Sums theorem

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