## Abstract

Let F be a family of convex sets in 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}⊂ F, for 1 ≤ i≤ 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≥ 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.

Original language | English |
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Pages (from-to) | 848-866 |

Number of pages | 19 |

Journal | Discrete and Computational Geometry |

Volume | 63 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jun 2020 |

## Keywords

- Colorful Helly-type theorems
- Convex sets
- Geometric transversals
- Line transversals
- Transversal numbers
- Weak epsilon-nets

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics