Almost all C₄-free graphs have fewer than (1 - ε)ex(n, C₄) edges

A graph is called H-free if it contains no copy o f H. Let ex(n, H) denote the Turán number for H, i.e., the maximum number of edges that an n-vertex H-free graph may have. An old result of Kleitman and Winston states that there are 2^o(ex(n,C4)) C₄-free graphs on n vertices. Füredi showed that almost all C₄-free graphs of order n have at least cex(n, C₄) edges for some positive constant c. We prove that there is a positive constant e such that almost all C ₄-free graphs have at most (1 - ε) ex(n, C₄) edges. This resolves a conjecture of Balogh, Bollobás, and Simonovits for the 4-cycle.