Balazs Keszegh

On the number of tangencies among 1-intersecting x-monotone curves

Abstract: 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.
Joint work with Eyal Ackerman.