  Consider the following stochastic graph process. We begin with the empty
graph on n vertices and add edges one at a time, where each edge is chosen
uniformly at random from the collection of potential edges that do not form
triangles when added to the graph. The process terminates at a maximal
traingle-free graph. Here we analyze the triangle-free process, determining the
likely order of magnitude of the number of edges in the final graph. As a
corollary we show that the triangle-free process is very likely to produce a
Ramsey R(3,t) graph; that is, our analysis of the triangle-free process gives a
new proof of the lower bound on R(3,t) previously established by Jeong Han Kim.
The techniques introduced extend to the K_4-free process thereby establishing a
small improvement in the best known lower bound on the Ramsey number R(4,t).
