Many T copies 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 = Km and χ(H) > m the behavior of ex(G(n, p),Km, H) depends strongly on the relation between p and m2(H) = maxH′⊂H,|V (H′)|′≥3 { e(H′)−1 v(H′)−2 } . When m2(H) > m2(Km) we prove that with high probability, depending on the value of p, either one can maintain almost all copies of Km, or it is asymptotically best to take a χ(H) − 1 partite subgraph of G(n, p). The transition between these two behaviors occurs at p = n−1/m2(H). When m2(H) < m2(Km) we show that the above cases still exist, however for δ > 0 small at p = n−1/m2(H)+δ one can typically still keep most of the copies of Km 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, i) where m2(G(k, i)) tends to (k+1)(k−2) 2(k−1) (< m2(Kk−1)) as i tends to infinity. This is tight for all values of k. Mathematics Subject Classification: 05C35, 05C80