A Crossing Lemma for Jordan Curves

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 $(1-o(1))n^2$.