Disjoint edges in topological graphs

Abstract

A topological graph G is a graph drawn in the plane so that its edges are represented by Jordan arcs. G is called simple, if any two edges have at most one point in common. It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k−8n) edges. The assumption that G is simple cannot be dropped: for every n, there exists a complete topological graph of n vertices, whose any two edges cross at most twice.