Random Ramsey graphs for the four-cycle
Discrete Mathematics
Tura´n's extremal problem in random graphs: forbidding even cycles
Journal of Combinatorial Theory Series B
Open problems of Paul Erd&ohuml;s in graph theory
Journal of Graph Theory
The number of graphs without forbidden subgraphs
Journal of Combinatorial Theory Series B
On the Number of Edges in Random Planar Graphs
Combinatorics, Probability and Computing
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
The typical structure of graphs without given excluded subgraphs
Random Structures & Algorithms
The fine structure of octahedron-free graphs
Journal of Combinatorial Theory Series B
The number of K m,m -free graphs
Combinatorica
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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.