# Natasha Dobrinen: The universal triangle-free graph has finite Ramsey degrees

We prove that the universal triangle-free graph has finite Ramsey degrees. The proof involves several new notions and constructions. First, a notion of strong triangle-free tree coding the universal triangle-free graph. Second, a proof of a Halpern-Lauchli-style theorem using the method of forcing but obtaining a result in ZFC. Third, new notions of strong similarity type and of subtree envelope for diagonal trees coding finite triangle-free graphs. Lastly, a diagonal tree coding the universal triangle-free graph along with a set of auxiliary witnessing nodes which together yield the Ramsey degrees for finite triangle-free graphs.