On short edges in complete topological graphs

TL;DR

This paper improves the upper bound on the minimum number of crossings for an edge in complete topological graphs, introducing new combinatorial bounds and a variant of a matching theorem related to VC-dimension.

Contribution

It establishes a tighter upper bound of O(n^{7/4}) for h(n) and introduces a novel variant of Chazelle and Welzl's matching theorem for set systems with bounded VC-dimension.

Findings

Proved h(n) = O(n^{7/4})

Developed a new variant of the matching theorem

Enhanced understanding of crossings in topological graphs

Abstract

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.