TY - GEN

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 - Publisher Copyright:
© Oswin Aichholzer, Martin Balko, Thomas Hackl, Jan Kynčl, Irene Parada, Manfred Scheucher.

PY - 2017/6/1

Y1 - 2017/6/1

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(n log4/5 n), 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 Pavel.

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(n log4/5 n), 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 Pavel.

KW - Birgit Vogtenhuber

KW - Empty k-gon

KW - Empty pentagon

KW - Erdos-Szekeres type problem

KW - K-hole

KW - Planar point set Valtr

UR - http://www.scopus.com/inward/record.url?scp=85029949132&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2017.8

DO - 10.4230/LIPIcs.SoCG.2017.8

M3 - Conference contribution

AN - SCOPUS:85029949132

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 81

EP - 816

BT - 33rd International Symposium on Computational Geometry, SoCG 2017

A2 - Katz, Matthew J.

A2 - Aronov, Boris

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 33rd International Symposium on Computational Geometry, SoCG 2017

Y2 - 4 July 2017 through 7 July 2017

ER -