Distinguishing subgroups of the rationals by their Ramsey properties

Ben Barber, Neil Hindman, Imre Leader, Dona Strauss

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

1 Citation (Scopus)
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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.
Original languageEnglish
Pages (from-to)93–104
Number of pages12
JournalJournal of Combinatorial Theory, Series A
Volume129
Early online date15 Oct 2014
DOIs
Publication statusPublished - 1 Jan 2015

Keywords

  • Partion regular
  • Rationals
  • Subgroups
  • Central sets

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