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A Turing degree a is said to be almost everywhere dominating if, for almost all X ∈ 2 ω with respect to the " fair coin " probability measure on 2 ω , and for all g : ω → ω Turing reducible to X, there exists f : ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other,… (More)

Motivated by Tukey classification problems and building on work in Part 1 [5], we develop a new hierarchy of topological Ramsey spaces Rα, α < ω 1. These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space [7], and for each α < ω 1 , R α+1 coming immediately after Rα in complexity. Associated with each Rα is an ultrafilter Uα, which… (More)

We study ultrafilters on ω 2 produced by forcing with the quotient of P(ω 2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only non-principal ultrafilters strictly below it in the Rudin-Keisler order are a single isomorphism class of selective ultrafilters. We further show… (More)

We characterize a class of topological Ramsey spaces such that each element R of the class induces a collection {R k } k<ω of projected spaces which have the property that every Baire set is Ramsey. Every projected space R k is a subspace of the corresponding space of length-k approximation sequences with the Tychonoff, equivalently metric, topol-ogy. This… (More)