For a free ultrafilter U on ω we study several cardinal characteristics which describe part of the combinatorial structure of U . We provide various consistency results; e.g. we show how to force simultaneously many characters and many π–characters. We also investigate two ideals on the Baire space ω naturally related to U and calculate cardinal coefficients of these ideals in terms of cardinal...

We provide an ultrafilter proof of the 2-dimensional Halpern–Läuchli Theorem in the following sense. If T0 and T1 are trees and T0 ⊗ T1 denotes their level product, we exhibit an ultrafilter U ∈ β(T0 ⊗ T1) so that every A ∈ U contains a subset of the form S0 ⊗ S1 for suitable strong subtrees of T0 and T1. We then discuss obstacles to extending our method of proof to higher dimensions.

There are two major types of ultrafilters, principal ultrafilters and non-principal ultrafilters. A principal ultrafilter is one that is generated by any single element a ∈ S, and has the form 〈a〉 = {A ⊆ S | a ∈ A}. However, these principal ultrafilters are not incredibly interesting, and will be used only as an example in this document. On the other hand, we have non-principal ultrafilters. We...

In [1], it was shown that the generic ultrafilter G2 forced by P(ω × ω)/(Fin ⊗ Fin) is neither maximum nor mimumum in the Tukey order of ultrafilters, but 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 all nonprincipal ultrafilters Tukey reducible to...

We introduce the relation of almost-reduction in an arbitrary topological Ramsey space R as a generalization of the relation of almostinclusion on N[∞]. This leads us to a type of ultrafilter U on the set of first approximations of the elements of R which corresponds to the well-known notion of selective ultrafilter on N. The relationship turns out to be rather exact in the sense that it permit...

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 Theorem canonizing equivalence relatio...

We are interested in generalizing part of the theory of ultrafilters on ω to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal. 0. Introduction A lot of knowledge has been accumulated on ultrafilters on ω. In the present paper we are interested in carrying out some of those arguments for ultrafilters on...

We use Shelah’s theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem 1. Suppose that λ is a singular cardinal, λ′ < λ, and the ultrafilter D is κ-decomposable for all regular cardinals κ with λ′ < κ < λ. Then D is either λ-decomposable, or λ-decomposable. Corollary 2. If λ is a singular cardinal, then an ultrafilter is (λ, λ)-regular if and only if it is e...

For a free ultrafilter U on ω we study several cardinal characteristics which describe part of the combinatorial structure of U . We provide various consistency results; e.g. we show how to force simultaneously many characters and many π–characters. We also investigate two ideals on the Baire space ωω naturally related to U and calculate cardinal coefficients of these ideals in terms of cardina...

Let 91 be a ~-complete ultrafilter on the measurable cardinal ~. Scott [ 1 3 ] proved V ¢ L by using 91 to take the ultrapower of V. Gaifman [ 2] considered iterated ultrapowers of V by cg to conclude even stronger results; for example, that L n ~(6o) is count-able. In this paper we discuss some new applications of iterated ultra-powers. In § § 1-4, we develop a straightforward generalization o...