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 : ù → ù Turing reducible to X , there… (More)

Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces Rα, α < ω1. These spaces form a natural hierarchy of complexity, R0… (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… (More)

Relative to a hyperstrong cardinal, it is consistent that measure one covering fails relative to HOD. In fact it is consistent that there is a superstrong cardinal and for every regular cardinal κ ,… (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… (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… (More)

We study ultrafilters on ω produced by forcing with the quotient of P(ω) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a Ppoint and that… (More)

This paper investigates when it is possible for a partial ordering P to forcePκ(ë) \ V to be stationary in V . It follows from a result of Gitik that whenever P adds a new real, then Pκ(ë) \ V 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… (More)