Let R be the set of all finite graphs G with the Ramsey property that
every coloring of the edges of G by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let G(n,p) be the random graph on n vertices with edge
probability p. We prove that there exists a function c^=c^(n) with
00, as n tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.