Nowhere dense Ramsey sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909728598130688 |
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| author | Rödl, Vojtěch Sales, Marcelo |
| author_facet | Rödl, Vojtěch Sales, Marcelo |
| contents | A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the area, a stronger ``density'' concept was considered in [J. Amer. Math. Soc. 3, 1--7, 1990]: If $S$ is a $d$-dimensional simplex, then for any $μ>0$ there is an integer $d:=d(S,μ)$ and finite configuration $X\subseteq \mathbb{R}^d$ such that any subconfiguration $Y\subseteq X$ with $|Y|\geq μ|X|$ contains a copy of $S$. Complementing this, here we show the existence of $μ:=μ(S)$ and of an infinite configuration $X\subseteq \mathbb{R}^{\infty}$ with the property that any finite coloring of $X$ yields a monochromatic copy of $S$, yet for any finite set of points $Y\subseteq X$ contains a subset $Z\subseteq Y$ of size $|Z|\geq μ|Y|$ without a copy of $S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nowhere dense Ramsey sets Rödl, Vojtěch Sales, Marcelo Combinatorics A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the area, a stronger ``density'' concept was considered in [J. Amer. Math. Soc. 3, 1--7, 1990]: If $S$ is a $d$-dimensional simplex, then for any $μ>0$ there is an integer $d:=d(S,μ)$ and finite configuration $X\subseteq \mathbb{R}^d$ such that any subconfiguration $Y\subseteq X$ with $|Y|\geq μ|X|$ contains a copy of $S$. Complementing this, here we show the existence of $μ:=μ(S)$ and of an infinite configuration $X\subseteq \mathbb{R}^{\infty}$ with the property that any finite coloring of $X$ yields a monochromatic copy of $S$, yet for any finite set of points $Y\subseteq X$ contains a subset $Z\subseteq Y$ of size $|Z|\geq μ|Y|$ without a copy of $S$. |
| title | Nowhere dense Ramsey sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.17137 |