From Ramsey degrees to Ramsey expansions via weak amalgamation

Abstract: Under no additional assumptions, in this paper we construct a Ramsey expansion for every category of finite objects with finite small Ramsey degrees. Our construction is based on the relationship between small Ramsey degrees, weak amalgamation and recent results about weak Fra\"iss\'e categories. Namely, generalizing the fact that every Ramsey class has amalgamation, we show that classes with finite Ramsey degrees have weak amalgamation. We then invoke the machinery of weak Fra\"iss\'e categories to perform the construction. This improves previous similar results where an analogous construction was carried out under the assumption that everything sits comfortably in a bigger class with enough infrastructure, and that in this wider context there is an ultrahomogeneous structure under whose umbrella the construction takes place.