Calculating with Ramsey degrees.

In this paper we thoroughly investigate the relationship between the small Ramsey degrees and the big Ramsey degrees in arbitrary 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 show how to derive a monotonicity property for small Ramsey degrees from the analogous fact for the big Ramsey degrees. We prove that small Ramsey degrees are subadditive and show that equality is enforced by the expansion property. 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. We also prove that the small Ramsey degrees enjoy a submultiplicative property. We do not know whether the analogous property holds for big Ramsey degrees.