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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 discuss the relationship between various weak distributive laws and games in Boolean algebras. In the first part we give some game characterizations for certain forms of Prikry's " hyper-weak distributive laws " , and in the second part we construct Suslin algebras in which neither player wins a certain hyper-weak distributivity game. We conclude that in… (More)