  A hereditary property of combinatorial structures is a collection of
structures (e.g. graphs, posets) which is closed under isomorphism, closed
under taking induced substructures (e.g. induced subgraphs), and contains
arbitrarily large structures. Given a property P, we write P_n for the
collection of distinct (i.e., non-isomorphic) structures in a property P with n
vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of
P. Also, we write P^n for the collection of distinct labelled structures in P
with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled
speed of P.
  The possible labelled speeds of a hereditary property of graphs have been
extensively studied, and the aim of this paper is to investigate the possible
speeds of other combinatorial structures, namely posets and oriented graphs.
More precisely, we show that (for sufficiently large n), the labelled speed of
a hereditary property of posets is either 1, or exactly a polynomial, or at
least 2^n - 1. We also show that there is an initial jump in the possible
unlabelled speeds of hereditary properties of posets, tournaments and directed
graphs, from bounded to linear speed, and give a sharp lower bound on the
possible linear speeds in each case.
