  We introduce a containment relation of hypergraphs which respects linear
orderings of vertices and investigate associated extremal functions. We extend,
by means of a more generally applicable theorem, the n.log n upper bound on the
ordered graph extremal function of F=({1,3}, {1,5}, {2,3}, {2,4}) due to Z.
Furedi to the n.(log n)^2.(loglog n)^3 upper bound in the hypergraph case. We
use Davenport-Schinzel sequences to derive almost linear upper bounds in terms
of the inverse Ackermann function. We obtain such upper bounds for the extremal
functions of forests consisting of stars whose all centers precede all leaves.
