K4-free subgraphs of random graphs revisited

In Combinatorica 17(2), 1997, Kohayakawa, Luczak and Rodl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemeredi's regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G(n,p). In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemeredi's regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K-4, thereby providing a conceptually simple proof for the main result in the paper cited above.