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. We also show that every complete $n$-vertex simple topological graph contains a noncrossing path on $\Omega(n^{1/9})$ vertices. This improves the previously best known bound of $(\log n)^{1 - o(1)}$ due to Aichholzer et al., and independently, the author and Zeng.

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至多与$h(n)$条其他边相交的边。2009年，Kyn\v{c}l和Valtr证明了$h(n) = O(n^2/\log^{1/4} n)$，并给出了下界构造$h(n) = \Omega(n^{3/2})$。本文证明了$h(n) = O(n^{7/4})$。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。此外，我们还证明了每个$n$个顶点的完全简单拓扑图都包含一条包含$\Omega(n^{1/9})$个顶点的非交叉路径。这改进了Aichholzer等人以及作者与Zeng独立先前得到的最佳下界$(\log n)^{1 - o(1)}$。