Big Ramsey degrees and topological dynamics

We consider Fräıssé structures whose objects have finite big Ramsey degree and ask what consequences this has for the dynamics of the automorphism group. Motivated by a theorem of D. Devlin about the partition properties of the rationals, we define the notion of a big Ramsey structure, a single structure which codes the big Ramsey degrees of a given Fräıssé structure. This in turn leads to the definition of a completion flow ; we show that if a Fräıssé structure admits a big Ramsey structure, then the automorphism group admits a unique universal completion flow. We also discuss the problem of when big Ramsey structures exist and explore connections to the notion of oscillation stability defined by Kechris, Pestov, and Todorčević [10].