By B. Bollobás (Eds.)
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We may repeat part (A) of the proof of Theorem 7 replacing e( ) by h( ) and c by c'. Then we may assume that G' = G", but have to decrease c': replace the original condition by condition h ( G " ) s c"n2. How we define the graph R k as in the beginning of (B) of the proof of Theorem 7. (B) We prove that if n is sufficiently large and R k 3 CJ,then G" 3 CJ(r),where r is fixed, but arbitrarily large. Exactly as in the proof of Theorem 7, we can prove that G " contains at least c , n J cycles CJ,where c , > O is a constant.
Exactly the same argument gives that if G" is a random subgraph of K, of size [en]' (or is obtained from K,, by choosing each edge with probability 2 c ) then G" will contain a K2(r, t ) for t = 2'c'n - nZi3with probability tending to one, but for t = 2'c'n + n213only with probability tending to 0. This shows that prohibiting the odd cycles results in an increase of the constant from 2'c' to 2 2 r - ' c rand that the main point of our theorem is that the same result can be obtained by prohibiting just one odd cycle.
The first deep theorem of this 29 30 B . Bolloh&, P. Simonovits, E. Szeheredi kind was proved by Erdos and Stone [ 8 ] in 1946. This theorem is the basis of the theory of extremal graphs without forbidden subgraphs (see [1, Ch. VI]). Considerable extensions of it were proved by Bollobas and Erdos  and by BollobSts, Erdos and Simonovits . For fixed r and t the extremal graphs EX(n, K,(2, r, t ) ) were studied by Erdos and Simonovits . Our first aim in this note is to describe EX(n, K J 2 , r, c n ) ) , where r s 2 and c > O .