Fractional discrete Helly for pairs in a family of boxes
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910877046800384 |
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| author | Eom, Taehyun Kim, Minki Lee, Eon |
| author_facet | Eom, Taehyun Kim, Minki Lee, Eon |
| contents | Given a point set $S$ in $\mathbb{R}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Soberón proved a fractional version of Halman's theorem for axis-parallel boxes, showing that every finite family $F$ of axis-parallel boxes in $\mathbb{R}^d$ with positive density of $S$-intersecting $(d+1)$-tuples contains an $S$-intersecting subfamily of size linear in $|F|$. We prove that qualitatively the same conclusion can be achieved if the density of $S$-intersecting pairs is sufficiently large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional discrete Helly for pairs in a family of boxes Eom, Taehyun Kim, Minki Lee, Eon Combinatorics 52A35 Given a point set $S$ in $\mathbb{R}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Soberón proved a fractional version of Halman's theorem for axis-parallel boxes, showing that every finite family $F$ of axis-parallel boxes in $\mathbb{R}^d$ with positive density of $S$-intersecting $(d+1)$-tuples contains an $S$-intersecting subfamily of size linear in $|F|$. We prove that qualitatively the same conclusion can be achieved if the density of $S$-intersecting pairs is sufficiently large. |
| title | Fractional discrete Helly for pairs in a family of boxes |
| topic | Combinatorics 52A35 |
| url | https://arxiv.org/abs/2503.11997 |