On edges crossing few other edges in simple topological complete graphs

We study the existence of edges having few crossings with the other edges in drawings of the complete graph (more precisely, in simple topological complete graphs). A topological graph  T=(V,E)T=(V,E) is a graph drawn in the plane with vertices represented by distinct points and edges represented by Jordan curves connecting the corresponding pairs of points (vertices), passing through no other vertices, and having the property that any intersection point of two edges is either a common end-point or a point where the two edges properly cross. A topological graph is simple if any two edges meet in at most one common point.Let h=h(n)h=h(n) be the smallest integer such that every simple topological complete graph on nn vertices contains an edge crossing at most hh other edges. We show that Ω(n3/2)≤h(n)≤O(n2/log1/4n)Ω(n3/2)≤h(n)≤O(n2/log1/4n). We also show that the analogous function on other surfaces (torus, Klein bottle) grows as cn2cn2.