The structure of almost all graphs in a hereditary property

A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n 7→ |Pn|, where Pn denotes the graphs of order n in P . It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then |Pn| = 2 2/2, where r = r(P) ∈ N is the so-called ‘colouring number’ of P . However, their results tell us very little about the structure of a typical graph G ∈ P . In this paper we describe the structure of almost every graph in a hereditary property of graphs, P . As a consequence, we derive essentially optimal bounds on the speed of P , improving the Alekseev-Bollobás-Thomason Theorem, and also generalizing results of Balogh, Bollobás and Simonovits.