Partition regularity with congruence conditions

Ben Barber, Imre Leader

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An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. Given an image partition regular matrix A, can we also insist that each variable xi is a multiple of some given di? This is a question of Hindman, Leader and Strauss. Our aim in this short note is to show that the answer is negative. As an application, we disprove a conjectured equivalence between the two main forms of partition regularity, namely image partition regularity and kernel partition regularity.
Original languageEnglish
Pages (from-to)293-297
Number of pages5
JournalJournal of Combinatorics
Issue number3
Publication statusPublished - 2013


  • partition regular systems
  • Ramsey theory


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