No Infinite $(p,q)$-Theorem for Piercing Compact Convex Sets with Lines in $\mathbb{R}^3$

Abstract

An infinite $(p,q)$-theorem, or an $(\aleph_0,q)$-theorem, involving two families $\mathcal{F}$ and $\mathcal{G}$ of sets, states that if in every infinite subset of $\mathcal{F}$, there are $q$ sets that are intersected by some set in $\mathcal{G}$, then there is a finite set $S_{\mathcal{F}}\subseteq\mathcal{G}$ such that for every $C\in\mathcal{F}$, there is a $B\in S_{\mathcal{F}}$ with $C\cap B\neq\emptyset$. We provide an example demonstrating that there is no $(\aleph_0,q)$-theorem for piercing compact convex sets in $\mathbb{R}^3$ with lines by constructing a family $\mathcal{F}$ of compact convex sets such that it does not have a finite line transversal, but for any $t\in\mathbb{N}$, every infinite subset of $\mathcal{F}$ contains $t$ sets that are pierced by a line.