Daniel McGinnis: A family of convex sets in the plane satisfying the ( 4 , 3 ) -property can be pierced by nine points.

A family of sets is said to have the $(p,q)$-property if for every $p$ sets, $q$ of them have a common point. It was shown by Alon and Kleitman that if $F$ is a finite family of convex sets in $R^d$ and $q\ge d+1$, then there is come constant $c_d(p,q)$ number of points that pierces each set in $F$. A problem of interest is to improve the bounds on the numbers $c_d(p,q)$. Here, we show that $c_2(4,3)\le 9$, which improves the previous upper bound of 13 by Gyárfás, Kleitman, and Tóth. The proof combines a topological argument and a geometric analysis.