Related papers: Tukey-idempotency and strong p-points
Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more and more uses in multiple areas of mathematics, such as combinatorics, dynamics, and algebra, among others. The purpose of this…
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…
Given a topological Ramsey space $(\mathcal R,\leq, r)$, we extend the notion of semiselective coideal to sets $\mathcal H\subseteq\mathcal R$ and study conditions for $\mathcal H$ that will enable us to make the structure $(\mathcal…
We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$…
Answering a question of Dobrinen and Todorcevic, we prove that below any stable ordered-union ultrafilter $\mathcal{U}$, there are exactly four nonprincipal Tukey classes: $[\mathcal{U}], [\mathcal{U}_{\operatorname{min}}],…
Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We…
We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space $\mathcal{R}$, are the notions of selective for $\mathcal{R}$ and Ramsey for $\mathcal{R}$ equivalent? Every topological Ramsey space…
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 extend the class of ultrafilters $U$ over countable sets for which $U\cdot U\equiv_T U$, extending several results from \cite{Dobrinen/Todorcevic11}. In particular, we prove that for each countable ordinal $\alpha\geq 2$, the generic…
We show that if the existence of a supercompact cardinal $\kappa$ with a weakly compact cardinal $\lambda$ above $\kappa$ is consistent, then the following are consistent as well (where $\mathfrak{t}(\kappa)$ and $\mathfrak{u}(\kappa)$ are…
We define several notions of a limit point on sequences with domain a barrier in $[\omega]^{<\omega}$ focusing on the two dimensional case $[\omega]^2$. By exploring some natural candidates, we show that countable compactness has a number…
We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the weak Ultrapower Axiom is not equivalent to the Ultrapower…
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 construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$-complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$. This…
In this paper, we study some new examples of ideals on $\omega$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order -- known as the…
We first introduce and investigate a new class of $T_0$ spaces -- strong R-spaces, which are stronger than both R-spaces and strongly well-filtered spaces. It is proved that any sup-complete poset equipped with the upper topology is a…
We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be…
The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of theorems proved in different fields: for…
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…
We prove that if $p$ is a selective ultrafilter then ${\mathbb Q}^{(\kappa)}$ has a $p$-compact group topology without non-trivial convergent sequences, for each infinite cardinal $\kappa =\kappa^\omega$. In particular, this gives the first…