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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 games G η 1 (κ) and G η <λ (κ) are played by two players in η +-complete and max(η + , λ)-complete Boolean algebras, respectively. For cardinals η, κ such that κ <η = η or κ <η = κ, the (η, κ)-distributive law holds in a Boolean algebra B iff Player 1 does not have a winning strategy in G η 1 (κ). Furthermore , for all cardinals κ, the (η,… (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)

- Natasha Dobrinen
- 2015

This paper concentrates on the existence of canonical cofinal maps of three types: continuous, generated by finitary monotone end-extension preserving maps, and generated by monotone finitary maps. The main theorems prove that every monotone cofinal map on an ultrafilter of a certain class of ultrafilters is actually canonical when restricted to some filter… (More)

- Natasha Dobrinen
- 2004

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)

We are grateful to B. Balcar for pointing out the following error in [1]: In Example 2 (and hence Example 1), if η > ω, then η regular and ♦ η + do not suffice to construct an η +-Suslin algebra. The original construction breaks down at the point where we " choose " the appropriate branches B t , since ♦ η + is not strong enough to guarantee their… (More)