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Infinite partition regular matrices

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Abstract

We consider infinite matrices with entries fromℤ (and only finitely many nonzero entries on any row). A matrixA is partition regular overℕ provided that, whenever the setℕ of positive integers is partitioned into finitely many classes there is a vector\(\overrightarrow x\) with entries inℤ such that all entries ofA \(A\overrightarrow x\) lie in the same cell of the partition. We show that, in marked contrast with the situation for finite matrices, there exists a finite partition ofℕ no cell of which contains solutions for all partition regular matrices and determine which of our pairs of matrices must always have solutions in the same cell of a partition.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany

    Walter A. Deuber

  2. Department of Mathematics, Howard University, 20059, Washington, D.C., USA

    Neil Hindman

  3. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, England

    Imre Leader

  4. Fachbereich Informatik, LS II, Universität Dortmund, D-44221, Dortmund, Germany

    Hanno Lefmann

Authors
  1. Walter A. Deuber
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  2. Neil Hindman
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  3. Imre Leader
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  4. Hanno Lefmann
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Deuber, W.A., Hindman, N., Leader, I. et al. Infinite partition regular matrices. Combinatorica 15, 333–355 (1995). https://doi.org/10.1007/BF01299740

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  • DOI: https://doi.org/10.1007/BF01299740

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