TY - JOUR

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

AU - Aichholzer, Oswin

AU - Balko, Martin

AU - Hackl, Thomas

AU - Kyncl, Jan

AU - Parada, Irene

AU - Scheucher, Manfred

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

N1 - 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.

PY - 2020/7

Y1 - 2020/7

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
4/5n), 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
4/5n), 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.

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

VL - 173

JO - Journal of Combinatorial Theory, Series A

JF - Journal of Combinatorial Theory, Series A

SN - 0097-3165

M1 - 105236

ER -