Hereditary and Monotone Properties of Graphs

Summary Given a hereditary graph property $$\mathcal{P}$$ let $${\mathcal{P}}^{n}$$ be the set of those graphs in $$\mathcal{P}$$ on the vertex set {1, …, n}. Define the constant c n by $$\vert {\mathcal{P}}^{n}\vert = {2}^{c_{n}\left ({ n \atop 2} \right )}$$ . 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 $$\mathcal{P}$$ . 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.