On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers

Keller, Chaya; Smorodinsky, Shakhar; Tardos, Gábor

doi:10.1137/1.9781611974782.148

Cited by 16 publications

(29 citation statements)

References 11 publications

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“…Since there is no known example of a family of convex sets in R 2 that satisfies the (4, 3)-property for which the piercing number exceeds 3, we make the following conjecture: Over the last few decades extensive research has been done to improve the Alon-Kleitman bounds, see e.g., [20,21,22]. For an excellent survey on the (p, q)-problem we refer the reader to [23].…”

Section: R D -Convex Societiesmentioning

confidence: 99%

Piercing numbers in approval voting

Zerbib

2019

Mathematical Social Sciences

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We survey a host of results from discrete geometry that have bearing on the analysis of geometric models of approval voting. Such models view the political spectrum as a geometric space, with geometric constraints on voter preferences. Results on piercing numbers then have a natural interpretation in voting theory, and we survey their implications for various classes of geometric constraints on voter approval sets.

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Section: R D -Convex Societiesmentioning

confidence: 99%

Piercing numbers in approval voting

Zerbib

2019

Mathematical Social Sciences

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“…The combinatorial and geometrical properties of families of sets in Euclidean space has been a thriving area of research in discrete geometry and combinatorics for many decades. It was initiated by the classical theorem of Helly [24], asserting that if F is a family of convex sets in R d in which every d + 1 members have nonempty intersection, then some point lies in every set in F. This landmark result led to the study of covering numbers (also sometimes called piercing numbers or hitting numbers) of families of sets, that is, the minimal number of points needed to "pierce" every set in a family given some local intersection property [5,16,21,29,28]. Helly's theorem is sharp in general, but improved bounds on covering numbers can be obtained if one restricts the convex sets in question.…”

Section: Introductionmentioning

confidence: 99%

The geometry and combinatorics of discrete line segment hypergraphs

Oliveros

O’Neill

Zerbib

2020

Discrete Mathematics

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An r-segment hypergraph H is a hypergraph whose edges consist of r consecutive integer points on line segments in R 2 . In this paper, we bound the chromatic number χ(H) and covering number τ (H) of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for r ≥ 3, the covering number τ (H) is at most (r − 1)ν(H), where ν(H) denotes the matching number of H. We prove our conjecture in the case where ν(H) = 1, and provide improved (in fact, optimal) bounds on τ (H) for r ≤ 5. We also provide sharp bounds on the chromatic number χ(H) in terms of r, and use them to prove two fractional versions of our conjecture. 1 arXiv:1807.04826v1 [math.CO]

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“…The most notable result of this kind is by Kleitman et al [7] who showed that HD 2 (4, 3) ≤ 13 (compared to the upper bound of 345 obtained in [1]). Recently, it was shown in [6] that HD d (p, q) ≤ p − q + 2 for all ε > 0, p ≥ p 0 (ε), and q > p…”

Section: Introductionmentioning

confidence: 99%

“…The best currently known upper bound that holds for all q, HD d (p, q) =Õ(p d· q−1 q−d ) (also shown in [6]), is apparently far from being tight.…”

Section: Introductionmentioning

confidence: 99%

On piercing numbers of families satisfying the (p,q) property

Keller

Smorodinsky

2018

Computational Geometry

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The Hadwiger-Debrunner number HD d (p, q) is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in R d that satisfies the (p, q) property. Hadwiger and Debrunner showed that HD d (p, q) ≥ p − q + 1 for all q, and equality is attained for q > d−1 d p+1. Almost tight upper bounds for HD d (p, q) for a 'sufficiently large' q were obtained recently using an enhancement of the celebrated Alon-Kleitman theorem, but no sharp upper bounds for a general q are known.In [9], Montejano and Soberón defined a refinement of the (p, q) property: F satisfies the (p, q) r property if among any p elements of F , at least r of the q-tuples intersect. They showed that HD d (p, q) r ≤ p − q + 1 holds for all r > p q − p+1−d q+1−d ; however, this is far from being tight.In this paper we present improved asymptotic upper bounds on HD d (p, q) r which hold when only a tiny portion of the q-tuples intersect. In particular, we show that for p, q sufficiently large, HD d (p, q) r ≤ p − q + 1 holds with r = 1 p q 2d p q . Our bound misses the known lower bound for the same piercing number by a factor of less than pq d .Our results use Kalai's Upper Bound Theorem for convex sets, along with the Hadwiger-Debrunner theorem and the recent improved upper bound on HD d (p, q) mentioned above.

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On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers

Cited by 16 publications

References 11 publications

Piercing numbers in approval voting

Piercing numbers in approval voting

The geometry and combinatorics of discrete line segment hypergraphs

On piercing numbers of families satisfying the (p,q) property

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