Let \crs(Kn) be the minimum number of crossings over all rectilinear drawings of the complete graph on n vertices on the plane. In this paper we prove that \crs(Kn)<0.380473(4n)+Θ(n3); improving thus on the previous best known upper bound. This is done by obtaining new rectilinear drawings of Kn for small values of n, and then using known constructions to obtain arbitrarily large good drawings from smaller ones. The "small" sets where found using a simple heuristic detailed in this paper.
@article{arxiv.1403.1288,
title = {Computational search of small point sets with small rectilinear crossing number},
author = {Ruy Fabila-Monroy and Jorge López},
journal= {arXiv preprint arXiv:1403.1288},
year = {2014}
}