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, Kyn\v{c}l and Valtr showed that $h(n) = O(n^2/\log^{1/4} n)$, and in the other direction, gave constructions showing that $h(n) = \Omega(n^{3/2})$. In this paper, we prove that $h(n) = O(n^{7/4})$. 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.

翻译：设$h(n)$为最小整数，使得每个$n$个顶点的完全简单拓扑图都包含一条与至多$h(n)$条其他边相交的边。2009年，Kyn\v{c}l和Valtr证明了$h(n) = O(n^2/\log^{1/4} n)$，另一方面，他们构造的例子表明$h(n) = \Omega(n^{3/2})$。在本文中，我们证明$h(n) = O(n^{7/4})$。在此过程中，我们建立了Chazelle和Welzl关于有界VC维集合系统的匹配定理的一个新变体，我们相信该变体具有独立的研究价值。