On the Size of Hereditary Classes of Graphs

A hereditary property of graphs is a class of graphs which is closed under taking induced subgraphs. For a hereditary property P, let Pn denote the set of P graphs on n labelled vertices. Clearly we have 0 ≤ |Pn| ≤ 2n(n − 1)/2, but much more can be said. Our main results show that the growth of |Pn| is far from arbitrary and that certain growth rates are impossible. For example, for no property P does one have |Pn| ∼ log n.