Local Ramsey Spaces in Matet Forcing Extensions and Finitely Many Near-Coherence Classes

We introduce Gowers--Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over $\omega$, and with this forcing we settle the long-standing problem of the spectrum of numbers near-coherence classes. We prove that for any finite $n \geq 1$, there is a forcing extension with exactly $n$ near-coherence classes of ultrafilters. For evaluating the new forcing, we prove a strengthening of Gowers's theorem on colourings of ${\rm Fin}_k$

The Filter Dichotomy Principle Does not Imply the Semifilter Trichotomy Principle

We answer Blass' question from 1989 of whether the inequality \gu < \gro is strictly stronger than the filter dichotomy principle affirmatively. We show that there is a forcing extension in which every non-meagre filter on $\omega$ is ultra by finite-to-one and the semifilter trichotomy does not hold. This trichotomy says: every semifilter is either meagre or comeagre or ultra by finite-to-one. The trichotomy is equivalent to the inequality \gu<\gro by work of Blass and Laflamme. Combinatorics of block sequences is used to establish forcing notions that preserve suitable properties of block sequences.Comment: This paper has been withdrawn by the author due to an error in Lemma 2.

Long low iterations

We try to control many cardinal characteristics by working with a notion of orthogonality between two families of forcings. We show that b^+<g is consisten

A Version of κ\kappa-Miller Forcing

Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa} = \kappa$ or just ${\rm cf}(\kappa) > \omega$, $2^{2^{<\kappa}}= 2^\kappa$, and $([\kappa]^\kappa, \supseteq)$ collapses $2^\kappa$ to $\omega$. We show under these assumptions the $\kappa$-Miller forcing with club many splitting nodes collapses $2^\kappa$ to $\omega$ and adds a $\kappa$-Cohen real

Specializing Aronszajn trees by countable approximations

We show that there are proper forcings based upon countable trees of creatures that specialize a given Aronszajn tree

On the cofinality of ultrapowers

All ultrafilters under consideration here are non-principal ultrafilters on the set omega of natural numbers. We are concerned with the possible cofinalities of ultrapowers of omega with respect to such ultrafilters. We show that no cardinal below the groupwise density number g can occur as such a cofinality and that at most one cardinal below the splitting number s can so occur. The proof for s, when combined with a result of Nyikos, gives the additional information that all P_{kappa}-point ultrafilters, for kappa greater than the bounding number b, are nearly coherent

On needed reals

Following Blass, we call a real a ``needed'' for a binary relation R on the reals if in every R-adequate set we find an element from which a is Turing computable. We show that every real needed for Cof(N) is hyperarithmetic. Replacing ``R-adequate'' by ``R-adequate with minimal cardinality'' we get related notion of being ``weakly needed''. We show that is is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetical reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999

The splitting number can be smaller than the matrix chaos number

Let chi be the minimum cardinal of a subset of 2^omega that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of creature forcing we show that s<chi is consistent. We thus answer a question by Vojtas. We give two kinds of models for the strict inequality. The first is the combination of an aleph_2-iteration of some proper forcing with adding aleph_1 random reals. The second kind of models is got by adding delta random reals to a model of MA_{< kappa} for some delta in [aleph_1,kappa). It was a conjecture of Blass that s=aleph_1<chi=kappa holds in such a model. For the analysis of the second model we again use the creature forcing from the first model

Many countable support iterations of proper forcings preserve Souslin trees

We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.Comment: 44 page

The relative consistency of g<cf(Sym(omega))

We prove the consistency result from the title. By forcing we construct a model of g=aleph_1, b=cf(Sym(omega))=aleph_2