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 D(S) of convex quadrilaterals determined by the points in S is at least 0.37553(4n) %2b O(n3). This in turn implies that the rectilineal crossing number cr̄(Kn) of the complete graph Kn is at least 0.37553 (4n) %2b O (n n), and that Sylvester%27s 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.
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) %2b O(n3). This in turn implies that the rectilineal crossing number cr̄(Kn) of the complete graph Kn is at least 0.37553 (4n) %2b 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.

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