Disjoint edges in complete topological graphs

It is shown that every complete $$n$$ -vertex simple topological graph has at $$\varOmega (n^{1/3})$$ pairwise disjoint edges, and these edges can be found in polynomial time. This proves a conjecture of Pach and Toth, which appears as Problem 5 from Chapter 9.5 in Research Problems in Discrete Geometry by Brass, Moser, and Pach.