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Many disjoint edges in topological graphs

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on $n$ vertices contains $Ω(n^{1-ε})$ pairwise disjoint edges for any $ε>0$. As a consequence, we show that every simple complete topological graph (drawn in the plane) with $n$ vertices contains $Ω(n^{\frac 12-ε})$ pairwise disjoint edges for any $ε>0$. This improves the previous lower bound of $Ω(n^\frac 13)$ by Suk which was reproved by Fulek and Ruiz-Vargas. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges.