Related papers: $\kappa$-Madness and Definability

Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not…

We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions in \cite{MR2354904} and \cite{MR2902230}. In particular, we show for inaccessible $\kappa$,…

We succeed to say something on the identities of (mu^+, mu) when mu>theta>cf(mu), mu strong limit theta--compact. This hopefully will help to prove the consistency of ``some pair (mu^+,mu) is not compact'', however, this has not been…

Let $x$ denote a Laver real over $L$. We prove that in $L[x]$ there is a $\Pi^1_1$ infinite mad family. Since $\Pi^1_1$ and $\Sigma^1_2$ sets are Laver measurable in $L[x]$, this shows that there are examples of well-behaved classical…

Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic.

Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardinal

Let $\kappa$ be a regular cardinal. Consider the Baire numbers of the spaces $(2^{\theta})_\kappa$ (functions from $\theta$ to 2 and the less than $\kappa$ topology) for various $\theta \geq \kappa$. Let l be the number of such different…

Let T be a complete, first-order theory in a finite or countable language having infinite models. Let I(T,kappa) be the number of isomorphism types of models of T of cardinality \kappa. We denote by \mu (respectively \hat\mu) the number of…

We show that under the Bounded Proper Forcing Axiom and an anti-large cardinal assumption, there is a $\mathbf{\Pi}^1_2$ MAD family.

Let $\mu < \kappa < \lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u (\kappa, \lambda)$ is regular, where $u (\kappa, \lambda)$ denotes the least size of a…

We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the…

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…

For an infinite cardinal mu, MAD(mu) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over mu. Erdos and Hechler proved the consistency of [mu in MAD(mu)] for a singular cardinal mu and asked if it was…

We introduce a covering notion depending on two cardinals, which we call $\mathcal O $-$ [ \mu, \lambda ]$-compactness, and which encompasses both pseudocompactness and many other generalizations of pseudocompactness. For Tychonoff spaces,…

We prove that: I. If $L$ is a $T_1$ space, $|L|>1$ and $d(L) \leq \kappa \geq \omega$, then there is a submaximal dense subspace $X$ of $L^{2^\kappa}$ such that $|X|=\Delta(X)=\kappa$; II. If $\frak{c}\leq\kappa=\kappa^\omega<\lambda$ and…

While maximal independent families can be constructed from ZFC via Zorn's lemma, the presence of a maximal $\sigma$-independent family already gives an inner model with a measurable cardinal, and Kunen has shown that from a measurable…

The structure of automorphism groups of $\kappa$-existentially closed groups are studied by Kaya-Kuzucuo\u{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^\kappa$ whenever…

We study conditions on automorphisms of Boolean algebras of the form $P(\lambda)/I_\kappa$ (where $\lambda$ is an uncountable cardinal and $I_\kappa$ is the ideal of sets of cardinality less than $\kappa$) which allow one to conclude that a…

We will give an overview of four families of cardinal characteristics defined on subspaces $\prod_{\alpha\in\kappa}b(\alpha)$ of the generalised Baire space ${}^\kappa\kappa$, where $\kappa$ is strongly inaccessible and…

In this paper we prove that if $\kappa$ is a singular cardinal with uncountable cofinality, then every power of a given topological space with precaliber $\kappa$ has precaliber $\kappa$ as well. Furthermore, if $\{X_\alpha :…