Many cliques in H-free subgraphs of random graphs

For two fixed graphs T and H let ex(G(n,p),T, H) be the random variable counting the maximum number of copies of T in an H-free subgraph of the random graph G(n,p). We show that for the case T = K-m and chi(H) > m the behavior of ex(G(n,p), K-m, H) depends strongly on the relation between p and m(2)(H) - MRXH' subset of, vertical bar V (H') vertical bar >= 3{e(H')-1/v(H')-2}. When m(2) (H) > m(2)(K-m) we prove that with high probability, depending on the value of p, either one can maintain almost all copies of K-m, or it is asymptotically best to take a chi(H) - 1 partite subgraph of G(n, p) . The transition between these two behaviors occurs at p = n(-1/m2(H)). When m(2)(H) < m(2)(K-m) we show that the above cases still exist, however for delta > 0 small at p = n(-1/m2(H)+delta) one can typically still keep most of the copies of K-m in an H-free subgraph of G(n,p). Thus, the transition between the two behaviors in this case occurs at some p significantly bigger than n(-1/m2(H)). To show that the second case is not redundant we present a construction which may be of independent interest. For each k >= 4 we construct a family of k-chromatic graphs G(k, epsilon(i)) where m(2) (G(k, epsilon(i))) tends to (k+1)(k-2)/2( k- 1) < m(2) (Kk-1) as i tends to infinity. This is tight for all values of k as for any k-chromatic graph G, m(2) (G) > (k+1)(k-2)/2(k-1).