The number of C₄-free graphs

Erdős, Frankl and Rödl showed that the number of graphs on n vertices with no copy of a non-bipartite graph H is 2(1+o(1))ex(n,H), whereas Morris and Saxton showed that this fails for C₆. Answering a question of Erdős, we prove that the number of C₄-free graphs on n vertices is 2(1+o(1))ex(n,C₄). The proof combines a modified hypergraph container method that treats pairs of vertices with large codegree separately, with a new bound on the number of matchings in a graph.