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Monochromatic sumsets in countable colourings of abelian groups

Volume 72 / 2024

Imre Leader, Kada Williams Bulletin Polish Acad. Sci. Math. 72 (2024), 97-102 MSC: Primary 05D10 DOI: 10.4064/ba240807-24-11 Published online: 18 December 2024

Abstract

Fernández-Bretón, Sarmiento and Vera showed that whenever a direct sum of sufficiently many copies of $\mathbb Z_4$, the cyclic group of order 4, is countably coloured there are arbitrarily large finite sets $X$ whose sumsets $X+X$ are monochromatic. They asked if the elements of order 4 are necessary, in the following strong sense: if $G$ is an abelian group having no elements of order 4, is it always the case that there is a countable colouring of $G$ for which there is not even a monochromatic sumset $X+X$ with $X$ of size 2? Our aim in this short note is to show that this is indeed the case.

Authors

  • Imre LeaderDepartment of Pure Mathematics and Mathematical Statistics
    Centre for Mathematical Sciences
    Cambridge CB3 0WB, UK
    e-mail
  • Kada WilliamsDepartment of Pure Mathematics and Mathematical Statistics
    Centre for Mathematical Sciences
    Cambridge CB3 0WB, UK
    e-mail

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