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

We prove that fractional Helly and $(p,q)$-theorems imply $({\mathrm{\aleph }}_0,q)$-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier $({\mathrm{\aleph }}_0,q)$-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 ${\mathrm{\aleph }}_0$ 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.