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

11 Citations (Scopus)

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

Funding

Aichholzer, Scheucher, and Vogtenhuber were partially supported by the ESF EUROCORES programme EuroGIGA – CRP ComPoSe, Austrian Science Fund (FWF): I648-N18 . Parada was supported by the Austrian Science Fund (FWF): W1230 . Balko and Valtr were partially supported by the grant GAUK 690214 . Balko, Kynčl, and Valtr were partially supported by the grant no. 18-19158S of the Czech Science Foundation (GAČR) and by the PRIMUS/17/SCI/3 project of Charles University . Balko and Kynčl were partially supported by Charles University project UNCE/SCI/004 . Hackl and Scheucher were partially supported by the Austrian Science Fund (FWF): P23629-N18 . Balko, Scheucher, and Valtr were partially supported by the ERC Advanced Research Grant no 267165 (DISCONV). Scheucher, Parada, and Vogtenhuber were partially supported within the collaborative DACH project Arrangements and Drawings, by grants DFG : FE 340/12-1 and FWF : I 3340-N35 , respectively. The research for this article was partially carried out in the course of the bilateral research project “Erdős–Szekeres type questions for point sets” between Graz and Prague, supported by the OEAD project CZ 18/2015 and project no. 7AMB15A T023 of the Ministry of Education of the Czech Republic .

Funders	Funder number
Seventh Framework Programme	267165
European Union's Horizon 2020 - Research and Innovation Framework Programme
Deutsche Forschungsgemeinschaft	I 3340-N35, FE 340/12-1
Austrian Science Fund	GAUK 690214, I648-N18, W1230, 18-19158S

Keywords

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

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10.1016/J.JCTA.2020.105236

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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. ",

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

year = "2020",

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AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

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N2 - 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.

AB - 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.

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A superlinear lower bound on the number of 5-holes

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