# Balázs Keszegh: The number of tangencies between two families of curves

## Description of video

 Date: 5/13/22 Speaker : Keszegh Balázs

## Keywords

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 $\mathrm{\Omega }\left({n}^{4/3}\right)$. 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 $\mathrm{\Theta }\left(n\mathrm{log}n\right)$, which improves a result by Pach, Suk, and Treml. Joint work with Dömötör Pálvölgyi.

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