Almost All $C_4$-Free Graphs Have Fewer than $(1-\varepsilon)\,\mathrm{ex}(n,C_4)$ Edges

Abstract

A graph is called H-free if it contains no copy of H. Let $\mathrm{ex}(n,H)$ 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 $2^{O(\mathrm{ex}(n,C_4))}$ $C_4$-free graphs on n vertices. Füredi showed that almost all $C_4$-free graphs of order n have at least $c\,\mathrm{ex}(n,C_4)$ edges for some positive constant c. We prove that there is a positive constant $\varepsilon$ such that almost all $C_4$-free graphs have at most $(1-\varepsilon)\,\mathrm{ex}(n,C_4)$ edges. This resolves a conjecture of Balogh, Bollobás, and Simonovits for the 4-cycle.