The number of graphs with large forbidden subgraphs

In this note, extending some results of Erdős, Frankl, Rödl, Alexeev, Bollobás and Thomason, we determine asymptotically the number of graphs which do not contain certain large subgraphs. In particular, we show that if H 1 , H 2 , … are graphs with | H n | = o ( log n ) and χ ( H n ) = r n + 1 , then the number S n of graphs of order n not containing H n satisfies log 2 S n = ( 1 − 1 / r n + o ( 1 ) ) ( n 2 ) . We also give a similar statement for forbidden induced subgraphs.