Extensions of discrete Helly theorems for boxes
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911848979234816 |
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| author | Edwards, Timothy Soberón, Pablo |
| author_facet | Edwards, Timothy Soberón, Pablo |
| contents | We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in $\mathbb{R}^d$. Halman's theorem says that, given a set $S$ in $\mathbb{R}^d$, if $F$ is a finite family of axis-parallel boxes such that the intersection of any $2d$ contains a point of $S$, then the intersection of $F$ contains a point of $S$. We prove colorful, fractional, and quantitative versions of Halman's theorem. For the fractional versions, it is enough to check that many $(d+1)$-tuples of the family contain points of $S$. Among the colorful versions we include variants where the coloring condition is replaced by an arbitrary matroid. Our results generalize beyond axis-parallel boxes to $H$-convex sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extensions of discrete Helly theorems for boxes Edwards, Timothy Soberón, Pablo Combinatorics 52A35 We prove extensions of Halman's discrete Helly theorem for axis-parallel boxes in $\mathbb{R}^d$. Halman's theorem says that, given a set $S$ in $\mathbb{R}^d$, if $F$ is a finite family of axis-parallel boxes such that the intersection of any $2d$ contains a point of $S$, then the intersection of $F$ contains a point of $S$. We prove colorful, fractional, and quantitative versions of Halman's theorem. For the fractional versions, it is enough to check that many $(d+1)$-tuples of the family contain points of $S$. Among the colorful versions we include variants where the coloring condition is replaced by an arbitrary matroid. Our results generalize beyond axis-parallel boxes to $H$-convex sets. |
| title | Extensions of discrete Helly theorems for boxes |
| topic | Combinatorics 52A35 |
| url | https://arxiv.org/abs/2404.14308 |