Monochromatic Sumsets in Countable Colourings of Abelian Groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910514351702016 |
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| author | Leader, Imre Williams, Kada |
| author_facet | Leader, Imre Williams, Kada |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monochromatic Sumsets in Countable Colourings of Abelian Groups Leader, Imre Williams, Kada Combinatorics 05D10 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. |
| title | Monochromatic Sumsets in Countable Colourings of Abelian Groups |
| topic | Combinatorics 05D10 |
| url | https://arxiv.org/abs/2407.03938 |