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(n4) + O(n3). This in turn implies that the rectilinear crossing number cr(Kn) of the complete graph Knis at least 0.37553(n4) + O(n3), and that Sylvester's Four Point Problem Constant is at least 0.37553. These improved bounds refine results recently obtained by Abrego and Fernandez-Merchant and by Lovasz, Vesztergombi, Wagner, and Welzl.