Improved Bounds for the Number of (≤ k)-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of K n

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 $0.37553\binom{n}{4} + O(n^3)$. This in turn implies that the rectilinear crossing number $\overline{\hbox{\rm cr}}(K_n)$ of the complete graph Kn is at least $0.37553\binom{n}{4} + O(n^3)$. These improved bounds refine results recently obtained by Abrego and Fernandez-Merchant, and by Lovasz, Vesztergombi, Wagner and Welzl.