The Ramsey theory of the universal homogeneous triangle-free graph Part II: Exact big Ramsey degrees

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.