Sharp thresholds for certain Ramsey properties of random graphs

In a series of papers culminating in [11] Rodl, Rucinski and others study t he 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 c opy 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 t he edge probability is at least Cp it almost surely has this property. We c omplement their results by showing that in certain cases, the multiplicativ e gap between upper and lower bound can be closed: There exists a function p(n) such that for every epsilon, a random graph with edge probability less than (1 - epsilon)p almost surely does not have the Ramsey property, where as when the edge probability is at least (1 + epsilon)p it almost surely ha s this property. However, this is an existence result only, since our metho d yields no information about the value of the function p(n). (C) 2000 John Wiley & Sons, Inc.