The surface $(p,q)$ piercing conjecture

Let $p$ and $q$ be integers with $p\geq q\geq 3$, let $S$ be a surface, and let $\mathcal{F}$ be a finite family of open connected subsets of $S$. The $(p,q)$ property means that among every $p$ members of $\mathcal{F}$, some $q$ have nonempty intersection; the piercing number is the minimum number of points of $S$ meeting every member of $\mathcal{F}$. Assume that the intersection of any members of $\mathcal{F}$ is empty or connected. Surface $(p,q)$ piercing conjecture. There exists an integer $C=C(p,q,S)$ such that, if $\mathcal{F}$ has the $(p,q)$ property, then its piercing number is at most $C$. The conjecture generalizes the stated theorem for particular surfaces; the paper notes that, assuming the complete-minor conjecture, earlier results would imply this assertion for arbitrary surfaces.