Typical Structure of Hereditary Graph Families. I. Apex-free Families

A family of graphs (Formula presented.) is hereditary if (Formula presented.) is closed under isomorphism and taking induced subgraphs. The speed of (Formula presented.) is the sequence (Formula presented.), where (Formula presented.) denotes the set of graphs in (Formula presented.) with the vertex set (Formula presented.). Alon et al. (2011) gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family (Formula presented.) there exist (Formula presented.) and an integer (Formula presented.) such that (Formula presented.) The main result of this paper gives a more precise description of typical structure for a restricted class of hereditary families. As a consequence we characterize hereditary families with the speed just above the threshold (Formula presented.), generalizing a result of Balogh and Butterfield (2011).