Enumeration of intersection graphs of x-monotone curves

A curve in the plane is x-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct 2^ (n^{4/3) } families, each consisting of n labelled x-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most 2^O (n^{3/2-) }, where >0 is a suitable constant. Our proof uses a new upper bound on the number of set systems of size m on a ground set of size n, with VC-dimension at most d. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.