K5-free subgraphs of random graphs

We consider a conjecture by Kohayakawa, Luczak, and Rodl, which, if true, would allow the application of Szemeredi's regularity lemma to show Turan-type results for all random graphs and any fixed graph H. We deal with 5-partite graphs such that each part contains n vertices, and any two parts form an e-regular pair with m edges. Here, two sets of vertices U and W of size n form an e-regular pair with m edges if all subsets U' ⊆ U and W' ⊆ W with [U'], [W'] ≥ en satisfy ||E(U', W')| - |U'| |W'|m/n2 ≤ e|U'| |W'|m/n2. We show that for each β > 0, there exists at most βm(n2m)10 such graphs that do not contain a K5 as a subgraph if m ≥ Cn5/3 for a constant C = C(β), e is sufficiently small, and n is sufficiently large. This proves the special case of the conjecture when H equals the complete graph on five vertices.