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 Rd 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 \(H{D_d}(p,d + 1)is\tilde O({p^{{d^2} + d}})\) .
We present several improved bounds: (i) For any \(q \geqslant d + 1,H{D_d}(p,d) = \tilde O({p^{d(\frac{{q - 1}}{{q - d}})}})\) . (ii) For q ≥ log p, \(H{D_d}(p,q) = \tilde O(p + {(p/q)^d})\) . (iii) For every ϵ > 0 there exists a p0 = p0(ϵ) such that for every p ≥ p0 and for every \(q \geqslant {p^{\frac{{d - 1}}{d} + \in }}\) 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 R2 with union complexity below a specific quadratic bound.
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A preliminary version of this paper was presented at the SODA’2017 conference.
Research partially supported by Grant 635/16 from the Israel Science Foundation.
Research partially supported by Grant 635/16 from the Israel Science Foundation. A part of this research was carried out during the authors’ visit at EPFL, supported by Swiss National Science Foundation grants 200020-162884 and 200021-165977.
Research partially supported by the “Lend¨ulet” project of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office, NKFIH, projects K-116769 and SNN-117879. A part of this research was carried out during the authors’ visit at EPFL, supported by Swiss National Science Foundation grants 200020-162884 and 200021-165977.
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Keller, C., Smorodinsky, S. & Tardos, G. Improved bounds on the Hadwiger–Debrunner numbers. Isr. J. Math. 225, 925–945 (2018). https://doi.org/10.1007/s11856-018-1685-1
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DOI: https://doi.org/10.1007/s11856-018-1685-1