A fractional Helly theorem for set systems with slowly growing homological shatter function
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | |
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| _version_ | 1866912135084244992 |
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| author | Bin, Marguerite |
| author_facet | Bin, Marguerite |
| contents | We study parameters of the convexity spaces associated with families of sets in $\mathbb{R}^d$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$. Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18605 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A fractional Helly theorem for set systems with slowly growing homological shatter function Bin, Marguerite Computational Geometry We study parameters of the convexity spaces associated with families of sets in $\mathbb{R}^d$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$. Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems. |
| title | A fractional Helly theorem for set systems with slowly growing homological shatter function |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2411.18605 |