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# Further consequences of the colorful Helly hypothesis

### Abstract:

Let $\F$ be a family of convex sets in $\reals^d$, 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 $d\geq 2$ 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 and Edgardo Roldan-Pensado.