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A Superlinear Lower Bound on the Number of 5-Holes Aichholzer, Oswin Balko, Martin Hackl, Thomas Kyncl, Jan Parada, Irene Scheucher, Manfred Valtr, Pavel Vogtenhuber, Birgit Erdös-Szekeres type problem k-hole empty k-gon empty pentagon planar point set 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 Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Oswin Aichholzer and Martin Balko and Thomas Hackl and Jan Kyncl and Irene Parada and Manfred Scheucher and Pavel Valtr and Birgit Vogtenhuber 2017 Is Part Of LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2017.8 urn:nbn:de:0030-drops-72008 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.8 eng https://creativecommons.org/licenses/by/3.0/legalcode