On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the planȩ so that any pair of them intersect and no three curves pass through the same point, is at least (1-o(1))n². We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R², so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Ω(nt log t/log log t).