Beyond the Richter-Thomassen Conjecture

摘要

If two closed Jordan curves in the plane have precisely one point in common, then it is called a touching point. All other intersection points are called crossing points. The main result of this paper is a Crossing Lemma for closed curves: In any family of n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, the number of crossing points exceeds the number of touching points by a factor of Ω((log log n)~(1/8)). As a corollary, we prove the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between any n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, is at least (1 - o(1))n~2.