Owings-like theorems for infinitely many colours or finite monochromatic sets
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909265011146752 |
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| author | Fernández-Bretón, David J. Rosales, Eliseo Sarmiento Vera, Germán |
| author_facet | Fernández-Bretón, David J. Rosales, Eliseo Sarmiento Vera, Germán |
| contents | Inspired by Owings's problem, we investigate whether, for a given an Abelian group $G$ and cardinal numbers $κ,θ$, every colouring $c:G\longrightarrowθ$ yields a subset $X\subseteq G$ with $|X|=κ$ such that $X+X$ is monochromatic. (Owings's problem asks this for $G=\mathbb Z$, $θ=2$ and $κ=\aleph_0$; this is known to be false for the same $G$ and $κ$ but $θ=3$.) We completely settle the question for $κ$ and $θ$ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for $κ$ and $θ$ both infinite (with a negative answer). Also, in the case where $θ$ is infinite but $κ$ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Owings-like theorems for infinitely many colours or finite monochromatic sets Fernández-Bretón, David J. Rosales, Eliseo Sarmiento Vera, Germán Logic Combinatorics Primary 03E05, Secondary 05D10, 05E16 Inspired by Owings's problem, we investigate whether, for a given an Abelian group $G$ and cardinal numbers $κ,θ$, every colouring $c:G\longrightarrowθ$ yields a subset $X\subseteq G$ with $|X|=κ$ such that $X+X$ is monochromatic. (Owings's problem asks this for $G=\mathbb Z$, $θ=2$ and $κ=\aleph_0$; this is known to be false for the same $G$ and $κ$ but $θ=3$.) We completely settle the question for $κ$ and $θ$ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for $κ$ and $θ$ both infinite (with a negative answer). Also, in the case where $θ$ is infinite but $κ$ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer. |
| title | Owings-like theorems for infinitely many colours or finite monochromatic sets |
| topic | Logic Combinatorics Primary 03E05, Secondary 05D10, 05E16 |
| url | https://arxiv.org/abs/2402.13124 |