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.
翻译:我们证明,两个家庭的成员(每个家庭由一美元组成,每个家庭由一美元构成的双向脱节曲线)之间的时间间隔可能与一美元(Omega,n ⁇ 4/3美元)一样大。我们从关于禁产的10美元-1美元的矩阵的猜测中可以看出,对于有双重原因的家庭来说,这一界限会非常明显。我们还表明,如果曲线必须是一美元,那么,最长时间间隔次数是一美元(n\log n),这改善了Pach、Suk和Treml的结果。 最后,我们还改进了已知的关于最多为美元(美元)的交叉曲线的家庭成员之间的时间间隔次数的界限。