Keszegh Balázs: Tangencies among 1-intersecting curves and an Erdős-Simonovits type theorem

According to a conjecture of Pach, there are O(n) tangent pairs among any family of n curves in which every pair of curves has precisely one common point and no three curves share a common point. This conjecture was proved for two special cases, however, for the general case the best upper bound was only O(n^{7/4}). This was also the best known bound on the number of tangencies in the relaxed case where every pair of curves has at most one common point. We improve the bounds for the former and latter cases to O(n^{3/2}) and O(n^{5/3}), respectively. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest. Joint work with Eyal Ackerman.