Related papers: Cascades, Order and Ultrafilters
We show the consistency of ZFC +''there is no NWD-ultrafilter on omega'', which means: for every non principle ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for…
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,…
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…
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…
In [1] the authors showed some basic properties of a pre-order that arose in combinatorial number theory, namely the finite embeddability between sets of natural numbers, and they presented its generalization to ultrafilters, which is…
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…
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in math.LO/0407498. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda-support iterations of…
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 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 continue the research of the relation $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ on the set $\beta {\mathbb{N}}$ of ultrafilters on ${\mathbb{N}}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as…
In this paper, we characterize the possible cofinalities of the least $\lambda$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $\delta$, that carries a $\lambda$-complete uniform ultrafilter, it is…
Given two infinite cardinals $\kappa$ and $\lambda$, we introduce and study the notion of a $\kappa$-barely independent family over $\lambda.$ We provide some conditions under which these types of families exist. In particular, we relate…
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 simple \(P_\lambda\)-point on a regular cardinal \(\kappa\) is a uniform ultrafilter on \(\kappa\) with a mod-bounded decreasing generating sequence of length \(\lambda\). We prove that if there is a simple $P_\lambda$-point ultrafilter…
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity…
In this paper, we investigate the poset $\mathbf{OF}(X)$ of free open filters on a given space $X$. In particular, we characterize spaces for which $\mathbf{OF}(X)$ is a lattice. For each $n\in\mathbb{N}$ we construct a scattered space $X$…
We define and study certain linear orders on chainable continua. Those orders depend on a sequence of chains obtained from definition of chainability and on a fixed non-principal ultrafilter on the set of natural numbers. An alternative…
We prove the consistency of a singular cardinal $\lambda$ with small value of the ultrafilter number $u_\lambda$, and arbitrarily large value of $2^\lambda$.
Motivated by a question of Isbell, we show that Jensen's Diamond Principle implies there is a non-P-point ultrafilter U on omega such that U, whether ordered by reverse inclusion or reverse inclusion mod finite, is not Tukey equivalent to…
We continue the study of the pseudo-intersection property with respect to an ideal introduced in \cite{TomNatasha2}. Our theory applies to the study of the Tukey types of general sums of ultrafilters, which, as evidenced by the results of…