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Infinite partition regular matrices
Authors:Walter A Deuber  Neil Hindman  Imre Leader  Hanno Lefmann
Institution:(1) Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany;(2) Department of Mathematics, Howard University, 20059 Washington, D.C., USA;(3) Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, CB2 1SB Cambridge, England;(4) Fachbereich Informatik, LS II, Universität Dortmund, D-44221 Dortmund, Germany
Abstract:We consider infinite matrices with entries fromZopf (and only finitely many nonzero entries on any row). A matrixA is partition regular overNopf provided that, whenever the setNopf of positive integers is partitioned into finitely many classes there is a vector 
$$\overrightarrow x$$
with entries inZopf 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 ofNopf 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.
Keywords:05 D 10
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