ar X iv : m at h / 07 02 35 2 v 1 [ m at h . C O ] 1 3 Fe b 20 07 HEREDITARY PROPERTIES OF ORDERED GRAPHS

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 order-preserving isomorphisms of the vertex set, and order-preserving induced subgraphs. If P is a hereditary property of ordered graphs, then Pn denotes the collection {G ∈ P : V (G) = [n]}, and the function n 7→ |Pn| is called the speed of P . The possible speeds of a hereditary property of labelled graphs have been extensively studied (see [9] and [11] for example), and more recently hereditary properties of other combinatorial structures, such as oriented graphs ([2], [7]), posets ([5], [16]), words ([4], [30]) and permutations ([22], [26]), have also attracted attention. Properties of ordered graphs generalize properties of both labelled graphs and permutations. In this paper we determine the possible speeds of a hereditary property of ordered graphs, up to the speed 2. In particular, we prove that there exists a jump from polynomial speed to speed Fn, the Fibonacci numbers, and that there exists an infinite sequence of subsequent jumps, from p(n)Fn,k to Fn,k+1 (where p(n) is a polynomial and Fn,k are the generalized Fibonacci numbers) converging to 2. Our results generalize a theorem of Kaiser and Klazar [22], who proved that the same jumps occur for hereditary properties of permutations.