Infinite partition regular matrices

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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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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Mathematics Subject Classification (1991)

  • 05 D 10