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

An infinite $(p,q)$-theorem, or an $({\mathrm{\aleph }}_0,q)$-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 $C\cap B\mathrm{\ne }\mathrm{\varnothing }$. We provide an example demonstrating that there is no $({\mathrm{\aleph }}_0,q)$-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.