TY - GEN

T1 - A new lower bound on Hadwiger-Debrunner numbers in the plane

AU - Keller, Chaya

AU - Smorodinsky, Shakhar

N1 - Publisher Copyright:
Copyright © 2020 by SIAM

PY - 2020

Y1 - 2020

N2 - A family of sets F is said to satisfy the (p, q) property if among any p sets in F some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p ≥ q ≥ d + 1 there exists an integer c = HDd(p, q), such that any finite family of convex sets in Rd that satisfies the (p, q) property can be pierced by at most c points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on HDd(p, q), known as 'the Hadwiger-Debrunner numbers', is still a major open problem in discrete and computational geometry. The best currently known upper bound on the Hadwiger-Debrunner numbers in the plane is O(p(1.5+δ)(1+ q− 1 2)) (for any δ > 0 and p ≥ q ≥ q0(δ)), obtained by combining results of Keller, Smorodinsky, and Tardos (SODA 2017) and of Rubin (FOCS 2018). The best lower bound is HD2(p, q) = Ω(pq log(pq )), obtained by Bukh, Matoušek and Nivasch more than 10 years ago. In this paper we improve the lower bound significantly by showing that HD2(p, q) ≥ p1+Ω(1/q). Furthermore, the bound is obtained by a family of lines and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers, which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the (p, q) problem to an old problem of Erdös on points in general position in the plane. We use a novel construction for Erdös' problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on HD2(p, 3). We then generalize the bound to HD2(p, q) for q ≥ 3.

AB - A family of sets F is said to satisfy the (p, q) property if among any p sets in F some q have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any p ≥ q ≥ d + 1 there exists an integer c = HDd(p, q), such that any finite family of convex sets in Rd that satisfies the (p, q) property can be pierced by at most c points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on HDd(p, q), known as 'the Hadwiger-Debrunner numbers', is still a major open problem in discrete and computational geometry. The best currently known upper bound on the Hadwiger-Debrunner numbers in the plane is O(p(1.5+δ)(1+ q− 1 2)) (for any δ > 0 and p ≥ q ≥ q0(δ)), obtained by combining results of Keller, Smorodinsky, and Tardos (SODA 2017) and of Rubin (FOCS 2018). The best lower bound is HD2(p, q) = Ω(pq log(pq )), obtained by Bukh, Matoušek and Nivasch more than 10 years ago. In this paper we improve the lower bound significantly by showing that HD2(p, q) ≥ p1+Ω(1/q). Furthermore, the bound is obtained by a family of lines and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers, which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the (p, q) problem to an old problem of Erdös on points in general position in the plane. We use a novel construction for Erdös' problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on HD2(p, 3). We then generalize the bound to HD2(p, q) for q ≥ 3.

UR - http://www.scopus.com/inward/record.url?scp=85084056918&partnerID=8YFLogxK

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AN - SCOPUS:85084056918

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1155

EP - 1169

BT - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

A2 - Chawla, Shuchi

T2 - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

Y2 - 5 January 2020 through 8 January 2020

ER -