## Abstract

A family F of sets satisfies the (p, q)-property if among every p members of F, some q can be pierced by a single point. The celebrated (p, q)-theorem of Alon and Kleitman asserts that for any p ? q ? d + 1, any family F of compact convex sets in R^{d} that satisfies the (p, q)-property can be pierced by a finite number c(p, q, d) of points. A similar theorem with respect to piercing by (d - 1)-dimensional flats, called (d - 1)-transversals, was obtained by Alon and Kalai. In this paper we prove the following result, which can be viewed as an (?0, k + 2)-theorem with respect to k-transversals: Let F be an infinite family of sets in R^{d} such that each A ? F contains a ball of radius r and is contained in a ball of radius R, and let 0 ? k < d. If among every ?0 elements of F, some k + 2 can be pierced by a k-dimensional flat, then F can be pierced by a finite number of k-dimensional flats. This is the first (p, q)-theorem in which the assumption is weakened to an (8, ·) assumption. Our proofs combine geometric and topological tools.

Original language | English |
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Title of host publication | 38th International Symposium on Computational Geometry, SoCG 2022 |

Editors | Xavier Goaoc, Michael Kerber |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772273 |

DOIs | |

State | Published - 1 Jun 2022 |

Event | 38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany Duration: 7 Jun 2022 → 10 Jun 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 224 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 38th International Symposium on Computational Geometry, SoCG 2022 |
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Country/Territory | Germany |

City | Berlin |

Period | 7/06/22 → 10/06/22 |

## Keywords

- (p,q)-theorem
- convexity
- infinite (p,q)-theorem
- k-transversal

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