Natasha Dobrinen

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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)
The Cohen algebra embeds as a complete subalgebra into three classic families of complete, atomless, c.c.c., non-measurable Boolean algebras; namely, the families of Argyros algebras and Galvin-Hajnal algebras, and the atomless part of each Gaifman algebra. It immediately follows that the weak (ω, ω)-distributive law fails everywhere in each of these(More)
The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ ++ ; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far(More)