Summary: THE STRUCTURE OF ALMOST ALL GRAPHS IN A HEREDITARY
NOGA ALON, J´OZSEF BALOGH, B´ELA BOLLOB´AS, AND ROBERT MORRIS
Abstract. 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 |Pn|, where Pn
denotes the graphs of order n in P. It was shown by Alekseev, and by Bollob´as 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-Bollob´as-Thomason Theorem, and also generalizing results
of Balogh, Bollob´as and Simonovits.
In this paper we shall describe the structure of almost every graph in an arbitrary
hereditary property of graphs, P. As a corollary, we shall obtain bounds on the speed of
P which improve those proved by Alekseev  and Bollob´as and Thomason [16, 17], and