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

翻译：设$\cC$为平面上一组曲线，满足$\cC$中任意三条曲线均不交于同一点，且$\cC$中每一对曲线恰好相交于一点，该交点或为横截交叉点或为切触点。János Pach 猜想$\cC$中相互切触的曲线对数量为$O(|\cC|)$。我们证明该猜想对$x$-单调曲线成立。