On the Number of Graphs Without Large Cliques

In 1976 Erdos, Kleitman, and Rothschild determined asymptotically the logarithm of the number of graphs without a clique of a fixed size $\ell$. In this note we extend their result to the case of forbidden cliques of increasing size. More precisely we prove that for $\ell_n \le (\log n)^{1/4}/2$ there are $2^{(1-1/(\ell_n-1))n^2/2+o(n^2/\ell_n)} K_{\ell_n}$-free graphs of order $n$. Our proof is based on the recent hypergraph container theorems of Saxton and Thomason and Balogh, Morris, and Samotij, in combination with a theorem of Lovasz and Simonovits.

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