Related papers: Ultrafilters on omega
We are interested in generalizing part of the theory of ultrafilters on omega to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
A set X which is a subset of the Cantor set has property (s) (Marczewski (Spzilrajn)) iff for every perfect set P there exists a perfect set Q contained in P such that Q is a subset of X or Q is disjoint from X. Suppose U is a nonprincipal…
We show the consistency of the set of regular cardinals which are the character of some ultrafilter on omega is not convex. We also deal with the set of pi chi-characters of ultrafilters on omega.
For a free ultrafilter U on omega 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…
An earlier paper, entitled "P-hierarchy on $\beta\omega$", investigated the relations between ordinal ultrafilters and the so-called P-hierarchy. This study is continued in the present paper and focuses on the aspects of characterization of…
We define a family of a (non-principal) ultrafilters on N which are, in a sense, far from P-points. We first under reasonable conditions, prove its existence. In a continuation we shall prove that such a point may exist while no P-point…
Under MA we prove that for the ideal $\cal I$ of thin sets on $\omega$ and for any ordinal $\gamma \leq \omega_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_{\gamma}$ of…
Following Baumgartner [J. Symb. Log. 60 (1995), no. 2], for an ideal $\mathcal{I}$ on $\omega$, we say that an ultrafilter $\mathcal{U}$ on $\omega$ is an $\mathcal{I}$-ultrafilter if for every function $f:\omega\to\omega$ there is $A\in…
We exhibit a forcing for producing a model with no nowhere dense ultrafilters that satisfies the full Sacks Property. By interleaving this forcing with other forcing notions, a model containing a $(2, {\aleph}_{0})$-selective ultrafilter,…
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…
We prove that \textsf{P}-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph_{\omega}% $-many random reals to a model of \textsf{CH. }These results improve and correct…
Under CH we prove that for any tall ideal $\cal I$ on $\omega$ and for any ordinal $\gamma \leq \omega_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_{\gamma}$ of P-hierarchy of…
We use Shelah's theory of possible cofinalities in order to solve a problem about ultrafilters. THEOREM. Suppose that $ \lambda $ is a singular cardinal, $ \lambda ' < \lambda $, and the ultrafilter $D$ is $ \kappa $-decomposable for all…
We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. Using that relation we prove (ZFC) that the class of strict $J_{\omega^\omega}$-ultrafilters, introduced by J. E. Baumgartner in…
We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of…
Various questions posed by P. Nyikos concerning ultrafilters on $\omega$ and chains in the partial order $(\omega,<^*)$ are answered. The main tool is the oracle chain condition and variations of it.
We introduce $\textit{Laver ultrafilters}$, namely ultrafilters $\mathcal{U}$ for which the associated Laver forcing $\mathbb{L}_{\mathcal{U}}$ has the Laver property. We give simple combinatorial characterisations of these ultrafilters,…
It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a…
We introduce the notion of a coherent $P$-ultrafilter on a complete ccc Boolean algebra, strenghtening the notion of a $P$-point on $\omega$, and show that these ultrafilters exist generically under ${\mathfrak c} = {\mathfrak d}$. This…
We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness…