The number of tangencies between two families of curves
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We prove that the number of tangencies between the members of two families, each of which consists of n pairwise disjoint curves, can be as large as Ω(n^4/3). We show that from a conjecture about forbidden 0-1 matrices it would follow that this bound is sharp for doubly-grounded families. We also show that if the curves are required to be x-monotone, then the maximum number of tangencies is Θ(nlog n), which improves a result by Pach, Suk, and Treml. Finally, we also improve the best known bound on the number of tangencies between the members of a family of at most t-intersecting curves.
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