On the number of graphs without large cliques

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