An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants - Aichholzer, Oswin; Duque, Frank; Fabila-Monroy, Ruy; García-Quintero, Oscar; Hidalgo-Toscano, Carlos (Journal of Graph Algorithms and Applications, 2020)
A drawing of a graph in the plane is pseudolinear if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines, that is, any pair of these curves crosses precisely once.A special case is rectilinear drawings where the edges of the graph are drawn as straight line segments. The rectilinear (pseudolinear) crossing numberof a graph is the minimum number of pairs of edges of the graph thatcross in any of its rectilinear (pseudolinear) drawings. In this paper we describean ongoing project to continuously obtain better asymptotic upperbounds on the rectilinear and pseudolinear crossing number of the complete graph $K_n$.