# 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

Publikation: Beitrag in einer FachzeitschriftArtikel

## 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 \$n)\$ and~\$O(n^2)\$, respectively. We show that \$h_5(n) = n4/5n)\$, 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 \$ into two subsets, each of size at least 5 and not in convex position, then \$ intersects the convex hull of some 5-hole in~\$P\$. The proof of this result is computer-assisted.
Originalsprache englisch 1-31 31 Journal of Combinatorial Theory / A https://doi.org/10.1016/j.jcta.2020.105236 Veröffentlicht - 2019

## Fields of Expertise

• Information, Communication & Computing