Abstract
We consider Fraïssé 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 Fraïssé structure. This in turn leads to the definition of a completion flow ; we show that if a Fraïssé 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.
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