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

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.