TY - JOUR

T1 - A superlinear lower bound on the number of 5-holes

AU - Aichholzer, Oswin

AU - Balko, Martin

AU - Hackl, Thomas

AU - Kynčl, Jan

AU - Parada, Irene

AU - Scheucher, Manfred

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

N1 - Funding Information:
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.
Funding Information:
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 .
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/7/1

Y1 - 2020/7/1

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 h5(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 h5(n) have been of order Ω(n) and O(n2), respectively. We show that h5(n)=Ω(nlog4/5n), obtaining the first superlinear lower bound on h5(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 h5(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 h5(n) have been of order Ω(n) and O(n2), respectively. We show that h5(n)=Ω(nlog4/5n), obtaining the first superlinear lower bound on h5(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.

KW - Empty k-gon

KW - Empty pentagon

KW - Erdös–Szekeres type problem

KW - Planar point set

KW - k-Hole

UR - http://www.scopus.com/inward/record.url?scp=85079878658&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2020.105236

DO - 10.1016/j.jcta.2020.105236

M3 - Article

AN - SCOPUS:85079878658

SN - 0097-3165

VL - 173

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

M1 - 105236

ER -