  In this paper we use the Klazar-Marcus-Tardos method to prove that if a
hereditary property of partitions P has super-exponential speed, then for every
k-permutation pi, P contains the partition of [2k] with parts {i, pi(i) + k},
where 1 <= i <= k. We also prove a similar jump, from exponential to factorial,
in the possible speeds of monotone properties of ordered graphs, and of
hereditary properties of ordered graphs not containing large complete, or
complete bipartite ordered graphs.
  Our results generalize the Stanley-Wilf Conjecture on the number of
n-permutations avoiding a fixed permutation, which was recently proved by the
combined results of Klazar and of Marcus and Tardos. Our main results follow
from a generalization to ordered hypergraphs of the theorem of Marcus and
Tardos.
