  An ordered graph is a graph together with a linear order on its vertices. A
hereditary property of ordered graphs is a collection of ordered graphs closed
under taking induced ordered subgraphs. If P is a property of ordered graphs,
then the function which counts the number of ordered graphs in P with exactly n
vertices is called the speed of P.
  In this paper we determine the possible speeds of a hereditary property of
ordered graphs, up to the speed 2^(n-1). In particular, we prove that there
exists a jump from polynomial speed to speed F(n), the Fibonacci numbers, and
that there exists an infinite sequence of subsequent jumps, from p(n)F(n,k) to
F(n,k+1) (where p(n) is a polynomial and F(n,k) are the generalized Fibonacci
numbers) converging to 2^(n-1). Our results generalize a theorem of Kaiser and
Klazar, who proved that the same jumps occur for hereditary properties of
permutations.
