A note on infinite versions of $(p,q)$-theorems
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910728577875968 |
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| author | Jung, Attila Pálvölgyi, Dömötör |
| author_facet | Jung, Attila Pálvölgyi, Dömötör |
| contents | We prove that fractional Helly and $(p,q)$-theorems imply $(\aleph_0,q)$-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier $(\aleph_0,q)$-theorems about geometric hypergraphs that were proved recently. Some of the corollaries are new results, for example, we prove that if $\mathcal{F}$ is an infinite family of convex compact sets in $\mathbb{R}^d$ and among every $\aleph_0$ of the sets some $d+1$ contain a point in their intersection with integer coordinates, then all the members of $\mathcal{F}$ can be hit with finitely many points with integer coordinates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04066 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on infinite versions of $(p,q)$-theorems Jung, Attila Pálvölgyi, Dömötör Combinatorics 52A35 We prove that fractional Helly and $(p,q)$-theorems imply $(\aleph_0,q)$-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier $(\aleph_0,q)$-theorems about geometric hypergraphs that were proved recently. Some of the corollaries are new results, for example, we prove that if $\mathcal{F}$ is an infinite family of convex compact sets in $\mathbb{R}^d$ and among every $\aleph_0$ of the sets some $d+1$ contain a point in their intersection with integer coordinates, then all the members of $\mathcal{F}$ can be hit with finitely many points with integer coordinates. |
| title | A note on infinite versions of $(p,q)$-theorems |
| topic | Combinatorics 52A35 |
| url | https://arxiv.org/abs/2412.04066 |