A Crossing Lemma for Families of Jordan Curves with a Bounded Intersection Number

TL;DR

This paper establishes new bounds on the number of touching and intersection points in families of Jordan curves with limited intersections, extending and improving previous results in geometric graph theory.

Contribution

It introduces a Crossing Lemma for contact graphs of Jordan curves with bounded intersection number, using the string separator theorem to derive tighter bounds.

Findings

Bound of O(n^{2 - 1/(3m+15)}) on touching pairs in m-intersecting families

Lower bound of Ω(T·(T/n)^{1/(9m+45)}) on intersection points given T touching pairs

Extension of previous bounds for pairwise intersecting curves to m-intersecting families

Abstract

A family of closed simple (i.e., Jordan) curves is $m$-intersecting if any pair of its curves have at most $m$ points of common intersection. We say that a pair of such curves touch if they intersect at a single point of common tangency. In this work we show that any $m$-intersecting family of $n$ Jordan curves in general position in the plane contains $O\left(n^{2-\frac{1}{3m+15}}\right)$ touching pairs Furthermore, we use the string separator theorem of Fox and Pach in order to establish the following Crossing Lemma for contact graphs of Jordan curves: Let $\Gamma$ be an $m$-intersecting family of closed Jordan curves in general position in the plane with exactly $T=\Omega(n)$ touching pairs of curves, then the curves of $\Gamma$ determine $\Omega\left(T\cdot\left(\frac{T}{n}\right)^{\frac{1}{9m+45}}\right)$ intersection points. This extends the similar bounds that were previously established by Salazar for the special case of pairwise intersecting (and $m$-intersecting) curves. Specializing to the case at hand, this substantially improves the bounds that were recently derived by Pach, Rubin and Tardos for arbitrary families of Jordan curves.