The number of tangencies between two families of curves (Q6081406)

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 \(\theta(n \log n)\), which improves a result by \textit{J. Pach} et al. [Comput. Geom. 45, No. 3, 131--138 (2012; Zbl 1243.52001)]. 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.