A Crossing Lemma for Families of Jordan Curves with a Bounded Intersection Number
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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(n^2-1/3m+15) touching pairs.[%s] Furthermore, we use the string separator theorem of Fox and Pach <cit.> 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. 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.
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