Related papers: The Higher Cicho\'n Diagram

Through careful analysis of an argument of Brooke-Taylor and Rosicky, we show that the powerful image of any accessible functor is closed under colimits of $\kappa$-chains, $\kappa$ a sufficiently large almost measurable cardinal. This…

In this paper we shall consider a couple of properties of $\sigma$-ideals and study relations between them. Namely we will prove that $\mathfrak{c}$-cc $\sigma$-ideals are tall and that the Weaker Smital Property implies that every Borel…

Exacting and ultraexacting cardinals are large cardinal numbers compatible with the Zermelo-Fraenkel axioms of set theory, including the Axiom of Choice. In contrast with standard large cardinal notions, their existence implies that the…

We solve a well--known problem in the theory of compact scattered spaces and superatomic boolean algebras by showing that, under GCH and for each regular cardinal $\kappa \geq \omega$, there is a poset $\mathcal P_\kappa$ preserving all…

We show that if the real line is the disjoint union of kappa meager sets such that every meager set is contained in a countable union of them, then kappa = omega_1. This answers a question addressed by J. Cichon. We also prove two theorems…

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $\lambda$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $\kappa$ is a tall…

We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $\kappa \leq \lambda$, where $\kappa$ is the successor of a regular cardinal, there is a rigid saturated ideal on…

We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$ and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of…

It is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate…

In 1984, Ditor asked two questions: (1) For each $n\in\omega$ and infinite cardinal $\kappa$, is there a join-semilattice of breadth $n+1$ and cardinality $\kappa^{+n}$ whose principal ideals have cardinality $< \kappa$? (2) For each $n \in…

We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $\kappa$,…

Let $\mathrm{cof}(\mu)=\mu$ and $\kappa$ be a supercompact cardinal with $\mu<\kappa$. Assume that there is an increasing and continuous sequence of cardinals $\langle\kappa_\xi\mid \xi<\mu\rangle$ with $\kappa_0:=\kappa$ and such that, for…

For each natural number $n$, let $C^{(n)}$ be the closed and unbounded proper class of ordinals $\alpha$ such that $V_\alpha$ is a $\Sigma_n$ elementary substructure of $V$. We say that $\kappa$ is a \emph{$C^{(n)}$-cardinal} if it is the…

The paper establishes several inequalities between cardinal characteristics of the continuum. In particular, it is shown that the partition splitting number is not larger than the uniformity of the meagre ideal; not all sets of reals having…

This paper deals with variety of problems in pcf theory and infinitary combinatorics. We look at normal filters and prc, measures of the size of [lambda]^{<kappa}, pcf-inaccessibility, entangled orders (and narrow Boolean Algebras),…

A stationary subset S of a regular uncountable cardinal kappa reflects fully at regular cardinals if for every stationary set T subseteq kappa of higher order consisting of regular cardinals there exists an alpha in T such that S cap alpha…

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let {\sf BC}$_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture.…

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded…

We continue our investigation =of Shelah's interpretability orders $\trianglelefteq^*_\kappa$ as well as the new orders $\trianglelefteq^\times_\kappa$. In particular, we give streamlined proofs of the existence of minimal unstable,…

We deal with the problem of preserving various versions of completeness in (< kappa) --support iterations of forcing notions, generalizing the case ``S --complete proper is preserved by CS iterations for a stationary co-stationary S…