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arXiv · 1803.06229

Further Consequences of the Colorful Helly Hypothesis

Leonardo Martínez-Sandoval·Edgardo Roldán-Pensado·Natan Rubin
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Abstract

Let $\mathcal{F}$ be a family of convex sets in ${\mathbb R}^d$, which are colored with $d+1$ colors. We say that $\mathcal{F}$ 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 Lov\'asz states that for any such colorful family $\mathcal{F}$ there is a color class $\mathcal{F}_i\subset \mathcal{F}$, 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 $\mathcal{F}$ can be crossed by $g(d)$ lines.

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Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, Natan Rubin. 2018-03-16. Further Consequences of the Colorful Helly Hypothesis. https://arxiv.org/abs/1803.06229

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