Sharp thresholds for certain Ramsey properties of random graphs

In a series of papers culminating in [11] Rödl, Ruciński and others study the thresholds of Ramsey properties of random graphs i.e. for a given graph H, when does a random graph almost surely have the property that for every coloring of its vertices (edges) in r colors there exists a monochromatic copy of H. They prove in many cases the existence of a function p(n, H) and two constants c(H) and C(H) such that a random graph with edge probability at most cp almost surely does not have this Ramsey property, whereas when the edge probability is at least Cp it almost surely has this property. We complement their results by showing that in certain cases, the multiplicative gap between upper and lower hound can be closed: There exists a function p(n) such that for every ε, a random graph with edge probability less than (1 - ε)p almost surely does not have the Ramsey property, whereas when the edge probability is at least (1 + ε)p it almost surely has this property. However, this is an existence result only, since our method yields no information about the value of the function p(n).