Natasha Dobrinen

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A Turing degree a is said to be almost everywhere dominating if, for almost all X ∈ 2 with respect to the “fair coin” probability measure on 2, 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, similarly(More)
The games G 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 1 (κ). Furthermore, for all cardinals κ, the (η,∞)-distributive law(More)
We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a p-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of p-points and selective ultrafilters. Results fall into three main categories: comparison to a(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 < ω1. These spaces form a natural hierarchy of complexity, R0 being the Ellentuck space [7], and for each α < ω1, Rα+1 coming immediately after Rα in complexity. Associated with each Rα is an ultrafilter Uα, which is(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 enough(More)
The generic ultrafilter G2 forced by P(ω × ω)/(Fin ⊗ Fin) was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters ([1]), but it was left open where exactly in the Tukey order it lies. We prove that G2 is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each k ≥ 2, the collection of(More)
Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This extends the Pudlak-Rödl(More)