Daniel McGinnis:

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

Abstract: 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 $\mathcal{F}$ is a finite family of convex sets in $\mathbb{R}^d$ and $q\geq d+1$, then there is come constant $c_d(p,q)$ number of points that pierces each set in $\mathcal{F}$. A problem of interest is to improve the bounds on the numbers $c_d(p,q). Here, we show that $c_2(4,3)\leq 9$, which improves the previous upper bound of 13 by Gyarfas, Kleitman, and Toth. The proof combines a topological argument and a geometric analysis.