Some model-theoretic remarks on structural Ramsey theory

We present one novel result and two novel proofs of previously known results in structural Ramsey theory. Regarding the former, we give a new characterization of the Ramsey property for a Fraïssé class in terms colorings of induced substructures of its generic model (or Fraïssé limit). This result is obtained as a corollary of a more general theorem in which we show that under relatively mild hypothesis, every expansion N of a given א0-categorical structure M induces at least one generic expansion of M (roughly in the sense of [4]) “constrained by N .” As for new proofs of old facts, we give a new proof the famous theorem of [5] asserting that a Fraïssé class K has the Ramsey property if and only if the automorphism group of its generic model is extremely amenable. Here, we use very explicitly model-theoretic techniques to show that that a Fraïssé classK has the Ramsey property if and only if the automorphism group of its generic model is extremely amenable relative to Stone spaces. Finally, we give a novel proof of the fact that the generic model of a Ramsey class always carries a 0-definable linear ordering; this new demonstration makes essential appeals to the Ramsey property’s role in constructing generalized indiscernible sequences and generalized Ehrenfeucht-Mostowski models.