 
Summary: THE RECTILINEAR CROSSING NUMBER OF Kn: CLOSING IN
(OR ARE WE?)
BERNARDO M. ŽABREGO, SILVIA FERNŽANDEZMERCHANT, AND GELASIO SALAZAR
Abstract. The calculation of the rectilinear crossing number of complete
graphs is an important open problem in combinatorial geometry, with impor
tant and fruitful connections to other classical problems. Our aim in this work
is to survey the body of knowledge around this parameter.
1. Introduction
In a rectilinear (or geometric) drawing of a graph G, the vertices of G are re
presented by points, and an edge joining two vertices is represented by the straight
segment joining the corresponding two points. Edges are allowed to cross, but an
edge cannot contain a vertex other than its endpoints. The rectilinear (or geometric)
crossing number cr(G) of a graph G is the minimum number of pairwise crossings
of edges in a rectilinear drawing of G in the plane.
1.1. The relevance of cr(Kn). As with every graph theory parameter, there is
a natural interest in calculating the rectilinear crossing number of certain families
of graphs, such as the complete bipartite graphs Km,n and the complete graphs
Kn. The estimation of cr(Kn) is of particular interest, since cr(Kn) equals the
minimum number (n) of convex quadrilaterals defined by n points in the plane
in general position; the problem of determining (n) belongs to a collection of
