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 ). This in turn implies that the rectilinear crossing number cr(Kn ) of the complete graph Kn is at least 0.37533 ), and that Sylvester's Four Point Problem Constant is at least 0.37533. These improved bounds refine results recently obtained by Abrego and Fernandez--Merchant, and by Lovasz, Vesztergombi, Wagner and Welzl.

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It is known that every generalized configuration with n points has at least 3 ` k+2 2 ´ ( ≤ k)–pseudoedges, and that this bound is tight for k ≤ n/3 − 1. Here we show that this bound is no longer tight for (any) k> n/3 − 1. As a corollary, we prove that the usual and the pseudolinear (and hence the rectilinear) crossing numbers of the complete graph Kn are different for every n ≥ 10. It has been noted that all known optimal rectilinear drawings of Kn share a triangular–like property, which we abstract into the concept of 3–decomposability. We give a lower bound for the crossing numbers of all pseudolinear drawings of Kn that satisfy this property. This bound coincides with the best general lower bound known for the rectilinear crossing number of Kn, established recently in a groundbreaking work by Aichholzer, García, Orden, and Ramos. We finally use these results to calculate the pseudolinear (which happen to coincide with the rectilinear) crossing numbers of Kn for n ≤ 12 and n = 15.

For n ≤ 27 we present exact values for the maximum number h(n) of halving lines and eh(n) of halving pseudolines, determined by n points in the plane. For this range of values of n we also present exact values of the rectilinear cr(n) and the pseudolinear ecr(n) crossing number of the complete

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Abstract. The calculation of the rectilinear crossing number of complete graphs is an important open problem in combinatorial geometry, with important and fruitful connections to other classical problems. Our aim in this work is to survey the body of knowledge around this parameter. 1.