Improved bounds for the number of (≤ k)-Sets, convex quadrilaterals, and the rectilinear crossing number of Kn

Abstract

We use circular sequences to give an improved lower bound on the minimum number of (≤ k)-sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number □ (S) of convex quadrilaterals determined by the points in S is at least 0.37553(4n) + O(n3). This in turn implies that the rectilinear crossing number ̄cr(Kn) of the complete graph Kn is at least 0.37553(4n) + O(n3). These improved bounds refine results recently obtained by Abrego and Fernández-Merchant, and by Lovász, Vesztergombi, Wagner and Welzl. © Springer-Verlag Berlin Heidelberg 2004.