Ramsey degrees: big v. small

Abstract: In this paper we investigate algebraic properties of big Ramsey degrees in categories satisfying some mild conditions. As the first nontrivial consequence of the generalization we advocate in this paper we prove that small Ramsey degrees are the minima of the corresponding big ones. We also prove that big Ramsey degrees are subadditive and show that equality is enforced by an abstract property of objects we refer to as self-similarity. Finally, we apply the abstract machinery developed in the paper to show that if a countable relational structure has finite big Ramsey degrees, then so do its quantifier-free reducts. In particular, it follows that the reducts of (Q, <), the random graph, the random tournament and (Q, <, 0) all have finite big Ramsey degrees.