The Number of Tangencies Between Two Families of Curves

Abstract

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}). 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 (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.