Finite big Ramsey degrees in universal structures

Abstract Big Ramsey degrees of finite structures are usually considered with respect to a Fraisse limit. The only available strategy to prove that a Fraisse limit supports finite big Ramsey degrees was suggested by Sauer in 2006 and relies on representing structures by binary trees and then invoking Milliken's Theorem. In this paper we consider structures which are not Fraisse limits, and still have the property that their finite substructures have finite big Ramsey degrees in them. For example, the class of all finite acyclic oriented graphs is not a Fraisse age, and yet we show that there is a countably infinite acyclic oriented graph in which every finite acyclic oriented graph has finite big Ramsey degree. Our approach to proving this and a few more statements of similar kind is based on a new strategy of transporting the property from one category of structures to another using the machinery of category theory.