The structure of almost all graphs in a hereditary property

AbstractA hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n↦|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(1−1/r+o(1))(n2), 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 generalising results of Balogh, Bollobás and Simonovits