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arXiv · 1304.5173

Partition regularity with congruence conditions

Ben Barber·Imre Leader
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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.

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Ben Barber, Imre Leader. 2013-12-19. Partition regularity with congruence conditions. https://doi.org/10.4310/joc.2013.v4.n3.a1

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