Let $F$ be a family of convex sets in $\mathbb{R}^d$, which are colored with $d+1$ colors. We say that $F$ satisfies the Colorful Helly Property if every rainbow selection of $d+1$ sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family $F$ there is a color class $F_i \subset F$, for $1 \leq i \leq d+1$, whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension $d$ there exist numbers $f(d)$ and $g(d)$ with the following property: either one can find an additional color class whose sets can be pierced by $f(d)$ points, or all the sets in $F$ can be crossed by $g(d)$ lines.
Joint work with Leonardo Martinez-Sandoval and Edgardo Roldan-Pensado.