Threshold Functions for Asymmetric Ramsey Properties Involving Cliques

Abstract

Consider the following problem: For given graphs G and F ₁,..., F _k , find a coloring of the edges of G with k colors such that G does not contain F _i in color i. For example, if every F _i is the path P ₃ 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 G $_{n,{\it p}}$ in the symmetric case when k is fixed and F ₁=...=F _k =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). Their proof was, however, non-constructive. 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. For this reason we refer to n ^− β as a threshold function.

In this paper we address the case when F ₁,..., F _k are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of G _n,p with p ≤bn ^− β for some constants b=b(F ₁,..., F _k , k) and β = β(F ₁,..., F _k ). Kohayakawa and Kreuter conjectured that $n^{-\beta(F_1,\dots, F_k)}$ is a threshold function in this case. This algorithm can be also adjusted to produce a valid k-coloring in the symmetric case.