  We investigate extremal functions ex_e(F,n) and ex_i(F,n) counting maximum
numbers of edges and maximum numbers of vertex-edge incidences in simple
hypergraphs H which have n vertices and do not contain a fixed hypergraph F;
the containment respects linear orderings of vertices. We determine both
functions exactly if F has only distinct singleton edges or if F is one of the
55 hypergraphs with at most four incidences (we give proofs only for six
cases). We prove some exact formulae and recurrences for the numbers of
hypergraphs, simple and all, with n incidences and derive rough logarithmic
asymptotics of these numbers. Identities analogous to Dobinski's formula for
Bell numbers are given.
