A Crossing Lemma for Jordan Curves

TL;DR

This paper establishes a lower bound on the number of intersection points for families of simple plane curves with many touching points, confirming a long-standing conjecture about intersections of Jordan curves.

Contribution

It proves a Crossing Lemma relating touching and intersection points for simple curves, confirming the conjecture that pairwise intersecting Jordan curves have nearly quadratic intersection points.

Findings

Number of intersection points grows faster than touching points when touching points are many.

Confirms the conjecture that pairwise intersecting Jordan curves have at least approximately n^2 intersections.

Provides a logarithmic lower bound on intersection points based on touching points.

Abstract

If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a {\em touching point}. The main result of this paper is a Crossing Lemma for simple curves: Let $X$ and $T$ stand for the sets of intersection points and touching points, respectively, in a family of $n$ simple curves in the plane, no three of which pass through the same point. If $|T|>cn$, for some fixed constant $c>0$, then we prove that $|X|=\Omega(|T|(\log\log(|T|/n))^{1/504})$. In particular, if $|T|/n\rightarrow\infty$, then the number of intersection points is much larger than the number of touching points. As a corollary, we confirm the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between $n$ pairwise intersecting simple closed (i.e., Jordan) curves in the plane, no three of which pass through the same point, is at least $(1-o(1))n^2$.