On the number of complete subgraphs and circuits contained in graphs
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and denote by KP the complete graph of p vertices . A well known theorem of TUR.áN [6] states that every 9(n ; m(n, p) + 1) contains a Kp and that this result is best possible. Thus in particular every á(2n ; n 2 + 1) contains a triangle. Denote by f„(p ; 1) the largest integer so that every W(n; m(n, p) + 1) contains at least fn(p ; l) distinct K p 's. RADEMACHER proved that fn(3 ; 1) = [n/2] and I proved [1] that there exists a constant 0 < c < z so that for every