On short edges in complete topological graphs

Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, Kynčl and Valtr showed that $h(n)=O(n^2\mathrm{/}{\log }^{1\mathrm{/}4}n)$, and in the other direction, gave constructions showing that $h(n)=\mathrm{\Omega }(n^{3\mathrm{/}2})$. In this paper, we prove that $h(n)=O(n^{7\mathrm{/}4})$. Along the way, we establish a new variant of Chazelle and Welzl’s matching theorem for set systems with bounded VC-dimension, which we believe to be of independent interest.