We give a new upper bound for the rectilinear crossing number of the complete geometric graph . We prove that by means of a new construction based on an iterative duplication strategy starting with a set having a certain structure of halving lines.

The of a drawing of a graph is the largest number of crossings on any edge of the drawing. In a of a graph, the vertices are points in the plane in general position and the edges are straight-line segments. The of the complete graph , denoted by , is . . .

The Harary–Hill Conjecture states that the number of crossings in any drawing of the complete graph KnKn in the plane is at least Z(n):=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋Z(n):=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋. In this paper, we settle the Harary–Hill conjecture for she . . .