Monochromatic Sumsets in Countable Colourings of Abelian Groups
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 there 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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.03938
- arXiv:
- arXiv:2407.03938
- Bibcode:
- 2024arXiv240703938L
- Keywords:
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- Mathematics - Combinatorics;
- 05D10