The Number of Tangencies Between Two Families of Curves

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 $$\Omega (n^{4/3})$$ Ω ( n 4 / 3 ) . We show that from a conjecture about forbidden 0–1 matrices it would follow that this bound is sharp for so-called doubly-grounded families. We also show that if the curves are required to be x -monotone, then the maximum number of tangencies is $$\Theta (n\log n)$$ Θ ( n log 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.