Disjoint Edges in Complete Topological Graphs

Abstract

It is shown that every complete n-vertex simple topological graph has at Ω (n1/3) pairwise disjoint edges, and these edges can be found in polynomial time. This proves a conjecture of Pach and Tóth, which appears as Problem 5 from Chapter 9. 5 in Research Problems in Discrete Geometry by Brass, Moser, and Pach. © 2012 Springer Science+Business Media New York.