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

TL;DR

This paper demonstrates that fractional Helly and $(p,q)$-theorems imply infinite $(eth_0,q)$-theorems in an abstract setting, with applications to geometric hypergraphs and new results on convex sets in Euclidean space.

Contribution

It establishes a general framework connecting fractional Helly and $(p,q)$-theorems to infinite versions, providing new and reproofed results in geometric hypergraph theory.

Findings

Reproves almost all earlier infinite $(eth_0,q)$-theorems in geometric hypergraphs.

Proves a new result on convex sets in $ eal^d$ with integer-coordinate intersection points.

Shows the implications of fractional Helly and $(p,q)$-theorems for infinite combinatorial structures.

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.