  Building on previous work of the author, for each finite triangle-free graph
$\mathbf{G}$, we determine the equivalence relation on the copies of
$\mathbf{G}$ inside the universal homogeneous triangle-free graph,
$\mathcal{H}_3$, with the smallest number of equivalence classes so that each
one of the classes persists in every isomorphic subcopy of $\mathcal{H}_3$.
This characterizes the exact big Ramsey degrees of $\mathcal{H}_3$. It follows
that the triangle-free Henson graph is a big Ramsey structure.
