Journal of Combinatorics

Volume 4 (2013)

Number 3

Partition regularity with congruence conditions

Pages: 293 – 297

DOI: http://dx.doi.org/10.4310/JOC.2013.v4.n3.a1

Authors

Ben Barber (Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, United Kingdom)

Imre Leader (Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, United Kingdom)

Abstract

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 $x_i$ is a multiple of some given $d_i$? 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.

Keywords

partition regular systems, Ramsey theory

2010 Mathematics Subject Classification

Primary 05D10. Secondary 03E02.

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