Bounds on piercing and line-piercing numbers in families of convex sets in the plane

Abstract

A family of sets has the $(p,q)$ property if among any $p$ members of it some $q$ intersect. It is shown that if a finite family of compact convex sets in ${\mathbb{R}}^2$ has the $(p+1,2)$ property then it is pierced by $\lfloor \frac{p}{2}\rfloor +1$ lines. A colorful version of this result is proved as well. As a corollary, the following is proved: Let $\F$ be a finite family of compact convex sets in the plane with no isolated sets, and let $\F'$ be the family of its pairwise intersections. If $\F$ has the $(p+1,2)$ property and $\F'$ has the $(r+1,2)$ property, then $\F$ is pierced by $(\lfloor \frac{r}{2}{\rfloor }^2+\lfloor \frac{r}{2}\rfloor )p$ points when $r\ge 2$, and by $p$ points otherwise. The proofs use the topological KKM theorem.