TY - JOUR

T1 - Improved bounds on the Hadwiger–Debrunner numbers

AU - Keller, Chaya

AU - Smorodinsky, Shakhar

AU - Tardos, Gábor

N1 - Funding Information:
∗ 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, sup-ported by Swiss National Science Foundation grants 200020-162884 and 200021-165977. †† Research partially supported by the “Lendület” project of the Hungarian Acad-emy 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 Foun-dation grants 200020-162884 and 200021-165977. Received December 1, 2016 and in revised form May 6, 2017
Publisher Copyright:
© 2018, Springer New York LLC. All rights reserved.

PY - 2018/4/18

Y1 - 2018/4/18

N2 - Let HDd(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 HDd(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HDd(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, HDd(p, q) = Õ (p+(p/q)d). (iii) For every ε > 0 there exists a p0 = p0(ε) such that for every p ≥ p0 and for every (image found) we have p − q + 1 ≤ HDd(p, q) ≤ p − q + 2. The latter is the first near tight estimate of HDd(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.

AB - Let HDd(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 HDd(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HDd(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, HDd(p, q) = Õ (p+(p/q)d). (iii) For every ε > 0 there exists a p0 = p0(ε) such that for every p ≥ p0 and for every (image found) we have p − q + 1 ≤ HDd(p, q) ≤ p − q + 2. The latter is the first near tight estimate of HDd(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.

UR - http://www.scopus.com/inward/record.url?scp=85043501022&partnerID=8YFLogxK

U2 - 10.1007/s11856-018-1685-1

DO - 10.1007/s11856-018-1685-1

M3 - Article

AN - SCOPUS:85043501022

VL - 225

SP - 925

EP - 945

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 2

ER -