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.
Categories: math.CO math.LO