According to a conjecture of Pach, there are $O(n)$ tangent pairs among any family of $n$ Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only $O(n^{7/4})$. This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has \emph{at most} one common point. We improve the bounds for the latter and former cases to $O(n^{5/3})$ and $O(n^{3/2})$, respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are \emph{$x$-monotone}, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is $Θ(n^{4/3})$. Without this last condition the number of tangencies is $O(n^{4/3}(\log n)^{1/3})$, improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest.

翻译：根据Pach的猜想，在任意包含n条Jordan弧的族中，若任意两条弧恰有一个公共点且任意三条弧无公共点，则切点对的数量为$O(n)$。该猜想已在两种特殊情形下得到证明，但在一般情形下，目前的最佳上界仅为$O(n^{7/4})$。即使在放宽条件下（要求任意两条弧至多有一个公共点），该上界仍是已知最佳结果。我们将放宽条件和原猜想条件下的上界分别改进至$O(n^{5/3})$和$O(n^{3/2})$。此外，我们还探讨了这些问题的若干变体，例如证明了若所有弧均为\emph{$x$-单调}曲线，每对弧至多相交一次，且其左端点位于同一条垂直线上，则最大切点数为$Θ(n^{4/3})$。若无最后一项条件，切点数上界为$O(n^{4/3}(\log n)^{1/3})$，该结果改进了Pach和Sharir先前给出的上界。在证明过程中，我们建立了一个图论定理，该定理推广了Erdős和Simonovits的结果，可能具有独立的理论意义。