We consider free amalgamation classes in a finite binary relational language. Such classes are determined by forbidding a set F of finite irreducible structures, for example, the class of finite triangle-free graphs. When the forbidden set F is finite, works of Dobrinen, Zucker, and Balko-Chodounský-Dobrinen-Hubička-Konečný-Vena-Zucker show that the class has finite big Ramsey degrees and provides an exact characterization of them. For F infinite, it was conjectured by BCDHKVZ that the class would not have finite big Ramsey degrees. In joint work with Joey Lakerdas-Gayle, we disprove this conjecture by giving an abstract necessary and sufficient condition for finite big Ramsey degrees and giving an example of an infinite F satisfying it. Along the way, we manage to pose several new questions.