New Lower Bounds for the Number of (<=k)-Edges and the Rectilinear Crossing Number of Kn

We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ bn−2 2 c the number of (≤ k)-edges is at least Ek(S) ≥ 3 ( k + 2 2 ) + k ∑ j=b3 c (3j − n + 3), which, for b3 c ≤ k ≤ 0.4864n, improves the previous best lower bound in [11]. As a main consequence, we obtain a new lower bound on the rectilinear crossing number of the complete graph or, in other words, on the minimum number of convex quadrilaterals determined by n points in the plane in general position. We show that the crossing number is at least ( 41 108 + ε )(n 4 ) + O(n) ≥ 0.379688 ( n 4 ) + O(n), which improves the previous bound of 0.37533 ( n 4 ) +O(n) in [11] and approaches the best known upper bound 0.380559 ( n 4 ) + Θ(n) in [2]. The proof is based on a result about the structure of sets attaining the rectilinear crossing number, for which we show that the convex hull is always a triangle. ∗Research on this paper was started while the first author was a visiting professor at the Departamento de Matemáticas, Universidad de Alcalá, Spain. †Research partially supported by the FWF (Austrian Fonds zur Förderung der Wissenschaftlichen Forschung) under grant S09205-N12, FSP Industrial Geometry ‡Research partially supported by grant MCYT TIC2002-01541. §Research partially supported by grants MTM2005-08618-C02-02 and S-0505/DPI/0235-02. ¶Research partially supported by grants TIC2003-08933-C02-01 and S-0505/DPI/0235-02.