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 l. In this note we extend their result to the case of forbidden cliques of increasing size. More precisely we prove that for l(n) <= (log n)(1/4)/2 there are 2((1-1/(ln - 1))n2/2+o(n2/ln))K(ln)-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.