Big Ramsey degrees and topological dynamics

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 [10]. Introduction, main definitions, and statements of theoremsConsider the statement of Ramsey's theorem: for every k, r < ω, we have ω → (ω) k r This "arrow notation" is shorthand for the following statement: for any coloring γ :[ω] k → r, there is an infinite S ⊆ ω so that |γ"([S] k )| = 1. We often say that S is monochromatic for γ. Ramsey's theorem can be generalized in many directions. Erdős and Rado [4] considered coloring finite tuples from larger cardinals while demanding larger monochromatic sets. Galvin and Prikry [7], and later Ellentuck [3], considered suitably definable colorings of the infinite subsets of ω. The generalization that we will consider in this paper is that of structural Ramsey theory. As a warmup, we will consider the structure Q, ≤ of rationals and their linear order. In what follows we will just write Q. Let us call a subset S ⊆ Q a dense linear order, or DLO for short, if the ordering on S inherited from Q is dense and contains no maximum or 2010 Mathematics Subject Classification. Primary: 22F50; Secondary: 03C15, 03E02, 03E75, 05D10, 37B20, 54D80, 54H20.

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