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(Kn) of the complete graph Kn 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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics