Asymmetric Ramsey properties of random graphs involving cliques

Consider the following problem: For given graphs G and F1,..., Fk, find a coloring of the edges of G with k colors such that G does not contain Fi in color i. Rödl and Rucinski studied this problem for the random graph Gn, p in the symmetric case when k is fixed and F1=...=Fk=F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p < bn-β for some constants b=b(F,k) and β = β(F). This result is essentially best possible because for p > Bn-β, where B=B(F, k) is a large constant, such an edge-coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n-β(F1,..., Fk) for arbitrary F1,..., Fk. In this paper we address the case when F1,..., Fk are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of Gn, p with p < bn-β for some constant b = b(F1,..., Fk). This algorithm also extends to the symmetric case. We also show that there exists a constant B = B(F1,..., Fk) such that for p > Bn-β, the random graph Gn, p a.a.s. does not have a valid k-edge-coloring provided the so-called KLR-conjecture holds.