Balázs Keszegh: 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 $\Omega \left(n^{4/3}\right)$. From a conjecture about $0$-$1$ matrices it would follow that if the families are doubly-grounded then this bound is sharp. We also show that if the curves are required to be $x$-monotone, then the maximum number of tangencies is $\Theta \left(n\log n\right)$, which improves a result by Pach, Suk, and Treml. Joint work with Dömötör Pálvölgyi.