Let $\cal C$ be a set of curves in the plane such that no three curves in $\cal C$ intersect at a single point and every pair of curves in $\cal C$ intersect at exactly one point which is either a crossing or a touching point. According to a conjecture of J\'anos Pach the number of pairs of curves in $\cal C$ that touch each other is $O(|{\cal C}|)$. We prove this conjecture for $x$-monotone curves.

翻译：设$\cal C$为平面上一组曲线，满足$\cal C$中任意三条曲线不交于同一点，且$\cal C$中任意两条曲线恰好相交于一个点，该点要么是交叉点要么是切触点。根据János Pach的一个猜想，$\cal C$中相互切触的曲线对数量为$O(|{\cal C}|)$。我们针对$x$-单调曲线证明了该猜想。