# Further Consequences of The Colorful Helly Hypothesis

## Natan Rubin, Ben-Gurion University

## September 12, 2017

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 LovĂˇsz
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.