The maximum number of tangencies among convex regions with a triangle-free intersection graph

Let t(C) be the number of tangent pairs among a set C of n Jordan regions in the plane. Pach et al. [Tangencies between families of disjoint regions in the plane, in Proceedings of the 26th ACM Symposium on Computational Geometry (SoCG 2010), Snowbird, June 2010, pp. 423-428] showed that if consists of convex bodies and its intersection graph is bipartite, then t(c) ≤ 4n - Θ(1), and, moreover, there are such sets that admit at least 3n - Θ(√n) tangencies. We close this gap and generalize their result by proving that the correct bound is 3n - Θ(1), even when the intersection graph of C is only assumed to be triangle-free.