Leírás
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. We show that any m-intersecting family of n Jordan curves in general position in the plane contains O(n ^(2− 1/ (3m+15))) 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 Γ be an m-intersecting family of closed Jordan curves in general position in the plane with exactly T = Ω(n) touching pairs of curves, then the curves of Γ determine Ω (T(T/ n)^(1/ 9m+45)) intersection points.
Bibliogaphy:
Maya Bechler-Speicher: A Crossing Lemma for Families of Jordan Curves with a Bounded Intersection Number, arXiv.