  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 $2^{\Omega(n^{4/3})}$ 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 $2^{O(n^{3/2-\varepsilon})}$, where
$\varepsilon>0$ 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.
