Improved Helly numbers of product sets
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909588633157632 |
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| author | Arun, Srinivas Dillon, Travis |
| author_facet | Arun, Srinivas Dillon, Travis |
| contents | A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it exists, such that every $k$-intersecting family contains a point of $S$ in its intersection. In this paper, we improve bounds on the Helly number of product sets of the form $A^d$ for various sets $A \subseteq \mathbb{R}$, including the ``exponential grid'' $A = \{α^n : n \in \mathbb{N}\}$ and sets $A\subseteq \mathbb{Z}$ defined by congruence relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07262 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Helly numbers of product sets Arun, Srinivas Dillon, Travis Combinatorics Metric Geometry A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it exists, such that every $k$-intersecting family contains a point of $S$ in its intersection. In this paper, we improve bounds on the Helly number of product sets of the form $A^d$ for various sets $A \subseteq \mathbb{R}$, including the ``exponential grid'' $A = \{α^n : n \in \mathbb{N}\}$ and sets $A\subseteq \mathbb{Z}$ defined by congruence relations. |
| title | Improved Helly numbers of product sets |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2409.07262 |