  The K_4-free process starts with the empty graph on n vertices and at each
step adds a new edge chosen uniformly at random from all remaining edges that
do not complete a copy of K_4. Let G be the random maximal K_4-free graph
obtained at the end of the process. We show that for some positive constant C,
with high probability as $n \to \infty$, the maximum degree in G is at most $C
n^{3/5}\sqrt[5]{\log n}$. This resolves a conjecture of Bohman and Keevash for
the K_4-free process and improves on previous bounds obtained by Bollob\'as and
Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash
this shows that with high probability G has $\Theta(n^{8/5}\sqrt[5]{\log n})$
edges and is `nearly regular', i.e., every vertex has degree
$\Theta(n^{3/5}\sqrt[5]{\log n})$. This answers a question of Erd\H{o}s, Suen
and Winkler for the K_4-free process. We furthermore deduce an additional
structural property: we show that whp the independence number of G is at least
$\Omega(n^{2/5}(\log n)^{4/5}/\log \log n)$, which matches an upper bound
obtained by Bohman up to a factor of $\Theta(\log \log n)$. Our analysis of the
K_4-free process also yields a new result in Ramsey theory: for a special case
of a well-studied function introduced by Erd\H{o}s and Rogers we slightly
improve the best known upper bound.
