  We consider the triangle-free process: given an integer n, start by taking a
uniformly random ordering of the edges of the complete n-vertex graph K_n.
Then, traverse the ordered edges and add each traversed edge to an (initially
empty) evolving graph - unless its addition creates a triangle. We study the
evolving graph at around the time where \Theta(n^{3/2 + \epsilon}) edges have
been traversed for any fixed \epsilon \in (0,10^{-10}). At that time and for
any fixed triangle-free graph F, we give an asymptotically tight estimation of
the expected number of copies of F in the evolving graph. For F that is
balanced and have density smaller than 2 (e.g., for F that is a cycle of length
at least 4), our argument also gives a tight concentration result for the
number of copies of F in the evolving graph. Our analysis combines Spencer's
original branching process approach for analysing the triangle-free process and
the semi-random method.
