Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2412.04066 |
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Inhaltsangabe:
- 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.