### Abstract

Let HD_{d}(p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in ℝ^{d} which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger– Debrunner conjecture, Alon and Kleitman proved that HD_{d}(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HD_{d}(p, d+1) is Õ(image found). We present several improved bounds: (i) For any q ≥ d+1, HDd(p, q) = Õ (image found). (ii) For q ≥ log p, HD_{d}(p, q) = Õ (p+(p/q)^{d}). (iii) For every ε > 0 there exists a p_{0} = p_{0}(ε) such that for every p ≥ p_{0} and for every (image found) we have p − q + 1 ≤ HD_{d}(p, q) ≤ p − q + 2. The latter is the first near tight estimate of HD_{d}(p, q) for an extended range of values of (p, q) since the 1957 Hadwiger–Debrunner theorem. We also prove a (p, 2)-theorem for families in ℝ^{2} with union complexity below a specific quadratic bound.

Original language | English |
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Pages (from-to) | 925-945 |

Number of pages | 21 |

Journal | Israel Journal of Mathematics |

Volume | 225 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 18 2018 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Israel Journal of Mathematics*,

*225*(2), 925-945. https://doi.org/10.1007/s11856-018-1685-1