A note on infinite versions of $(p,q)$-theorems

Abstract

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.

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