A note on infinite versions of $(p,q)$-theorems
| Autor: | Jung, Attila, Pálvölgyi, Dömötör |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: A previous version of this work appeared as part of arXiv:2311.15646 (v1 and v2). The main theorem is included there, but the current version provides a clearer presentation and additional applications |
| Databáze: | arXiv |
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