In this talk, we shall survey a large body of enumerative results in the context
of \(H\)-free graphs, that is, graphs not containing a copy of a fixed graph \(H\) as a
subgraph. In particular, we shall show how the recent ``hypergraph containers''
theorem of Balogh, Morris, and the speaker, proved independently by Saxton and
Thomason, allows one to derive (for each \(H\)) an approximate structural description
of a typical (random) \(H\)-free graph with a given number of vertices and edges. In
several interesting cases, such as when \(H\) is a clique, these approximate structural
descriptions may be made precise. We shall also mention several challenging open
questions.
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