Hereditary and monotone properties of graphs

Abstract

Given a hereditary graph property P let Pn be the set of those graphs in P on the vertex set (1, …, n). Define the constant c n by |Pn|=2cn(n2). We show that the limit lim n → ∞ c n always exists and equals 1 − 1 ∕ r, where r is a positive integer which can be described explicitly in terms of P. This result, obtained independently by Alekseev, extends considerably one of Erdős, Frankl and Rödl concerning principal monotone properties and one of Prömel and Steger concerning principal hereditary properties.