Abstract: A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed (i.e., Jordan) curves in the plane which are in general position and any pair of them intersect, is at least (2-o(1))n. We establish an even stronger form of the above conjecture, which states that the overall number of proper intersection points must exceed, in asymptotic terms, the number of the touching pairs of curves. Joint work with Janos Pach and Gabor Tardos