The number of tangencies between two families of curves

TL;DR

This paper investigates the maximum number of tangencies between two families of curves, establishing bounds that depend on the curve properties and connecting the problem to forbidden matrix conjectures.

Contribution

It provides new bounds on tangencies for disjoint and $x$-monotone curves, and links the problem to conjectures in forbidden matrix theory.

Findings

Maximum tangencies between two families of disjoint curves is Ω(n^{4/3})

Maximum tangencies for $x$-monotone curves is Θ(n log n)

Improved bounds on tangencies for t-intersecting 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 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 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.