Threshold Functions for Asymmetric Ramsey Properties 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. For example, if every Fi is the path P3 on 3 vertices, then we are looking for a proper k-edge-coloring of G, i.e., a coloring of the edges of G with no pair of edges of the same color incident to the same vertex. Rödl and Ruciński 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 for some constants b = b(F, k) and β = β(F ). Their proof was, however, non-constructive. This result is essentially best possible because for p ≥ Bn , where B = B(F, k) is a large constant, such an edge-coloring does not exist. For this reason we refer to n as a threshold function. 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 constants b = b(F1, . . . , Fk, k) and β = β(F1, . . . , Fk). Kohayakawa and Kreuter conjectured that n −β(F1,...,Fk) is a threshold function in this case. This algorithm can be also adjusted to produce a valid k-coloring in the symmetric case.