A superlinear lower bound on the number of 5-holes

Oswin Aichholzer; Martin Balko; Thomas Hackl; Jan Kyncl; Irene Parada; Manfred Scheucher; Pavel Valtr; Birgit Vogtenhuber

doi:10.1016/J.JCTA.2020.105236

A superlinear lower bound on the number of 5-holes

Oswin Aichholzer, Martin Balko, Thomas Hackl, Jan Kyncl, Irene Parada, Manfred Scheucher (Corresponding author), Pavel Valtr, Birgit Vogtenhuber

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h ₅(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h ₅(n) have been of order Ω(n) and O(n ²), respectively. We show that h ₅(n)=Ω(nlog ^4/5⁡n), obtaining the first superlinear lower bound on h ₅(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line ℓ into two subsets, each of size at least 5 and not in convex position, then ℓ intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted.

Original language	English
Article number	105236
Number of pages	31
Journal	Journal of Combinatorial Theory, Series A
Volume	173
DOIs	https://doi.org/10.1016/J.JCTA.2020.105236
Publication status	Published - Jul 2020
Externally published	Yes

Bibliographical note

DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

Keywords

Empty k-gon
Empty pentagon
Erdös–Szekeres type problem
Planar point set
k-Hole

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title = "A superlinear lower bound on the number of 5-holes",

abstract = "Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h 5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h 5(n) have been of order Ω(n) and O(n 2), respectively. We show that h 5(n)=Ω(nlog 4/5⁡n), obtaining the first superlinear lower bound on h 5(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line ℓ into two subsets, each of size at least 5 and not in convex position, then ℓ intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted. ",

keywords = "Empty k-gon, Empty pentagon, Erd{\"o}s–Szekeres type problem, Planar point set, k-Hole",

author = "Oswin Aichholzer and Martin Balko and Thomas Hackl and Jan Kyncl and Irene Parada and Manfred Scheucher and Pavel Valtr and Birgit Vogtenhuber",

note = "DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.",

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AU - Scheucher, Manfred

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

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