Kőnig = Ramsey, A compactness lemma for Ramsey categories

We prove a new characterization of the Ramsey property of categories in terms of a generalized form of Kőnig’s tree lemma. Afterwards, we discuss its applications to structural Ramsey theory. In particular, we provide a new proof of the existence and uniqueness of minimal Ramsey expansions and a new transfer theorem which uses Grothendieck opfibrations and unifies several known Ramsey transfers.