  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.
