  The areas of Ramsey theory and random graphs have been closely linked ever
since Erd\H{o}s' famous proof in 1947 that the 'diagonal' Ramsey numbers $R(k)$
grow exponentially in $k$. In the early 1990s, the triangle-free process was
introduced as a model which might potentially provide good lower bounds for the
'off-diagonal' Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are
introduced one-by-one at random and added to the graph if they do not create a
triangle; the resulting final (random) graph is denoted $G_{n,\triangle}$. In
2009, Bohman succeeded in following this process for a positive fraction of its
duration, and thus obtained a second proof of Kim's celebrated result that
$R(3,k) = \Theta \big( k^2 / \log k \big)$.
  In this paper we improve the results of both Bohman and Kim, and follow the
triangle-free process all the way to its asymptotic end. In particular, we
shall prove that $$e\big( G_{n,\triangle} \big) \,=\, \left(
\frac{1}{2\sqrt{2}} + o(1) \right) n^{3/2} \sqrt{\log n },$$ with high
probability as $n \to \infty$. We also obtain several pseudorandom properties
of $G_{n,\triangle}$, and use them to bound its independence number, which
gives as an immediate corollary $$R(3,k) \, \ge \, \left( \frac{1}{4} - o(1)
\right) \frac{k^2}{\log k}.$$ This significantly improves Kim's lower bound,
and is within a factor of $4 + o(1)$ of the best known upper bound, proved by
Shearer over 25 years ago.
