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It is shown that every complete $n$-vertex simple topological graph has at least Ω(n1/3) 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.

Disjoint edges in complete topological graphs