k-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn

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 0.37533 ` n 4 ́ + O(n). This in turn implies that the rectilinear crossing number cr(Kn) of the complete graph Kn is at least 0.37533 ` n 4 ́ + O(n), and that Sylvester’s Four Point Problem Constant is at least 0.37533. These improved bounds refine results recently obtained by Ábrego and Fernández–Merchant, and by Lovász, Vesztergombi, Wagner and Welzl.