k-Sets, convex quadrilaterals, and the rectilinear crossing number of K_n*

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 D(S) of convex quadrilaterals determined by the points in S is at least 0.37553(₄ⁿ) + O(n³). This in turn implies that the rectilineal crossing number cr̄(K_n) of the complete graph K_n is at least 0.37553 (₄ⁿ) + O (n ⁿ), and that Sylvester's Four Point Problem Constant is at least 0.37553. These improved bounds refine results recently obtained by Ábrego and Fernández-Merchant and by Lovász, Vesztergombi, Wagner, and Welzl.