Lower Bounds for the Number of Small Convex k-Holes.
Oswin Aichholzer, Ruy Fabila‐Monroy, Thomas Hackl, Clemens Huemer, Alexander Pilz, Birgit Vogtenhuber · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 2012
Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds on h(k)(n), for 3 = n(2) - 32n/7 + 22/7, h(4)(n) >= n(2)/2 - 9n/4 - o(n), and h(5)(n) >= 3n/4 - o(n). We further settle several questions on sets of 12 points posed by Dehnhardt in 1987. (C) 2013 Elsevier B.V. All rights reserved.