Related papers: Piece selection and cardinal arithmetic

We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH…

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa,\lambda$ are infinite cardinals such that $\kappa^{+++} \leq \lambda$, $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}= \kappa^+$, and…

We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if $\kappa \leq \lambda$, then $$\sup_{|A| = \lambda} |S^\kappa(A)| =…

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 $\kappa$,…

For finite semidistributive lattices the map $\kappa$ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the $\kappa$-map in the context of torsion classes. It…

In this paper we investigate more characterizations and applications of $\delta$-strongly compact cardinals. We show that, for a cardinal $\kappa$ the following are equivalent: (1) $\kappa$ is $\delta$-strongly compact, (2) For every…

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 consider the following dichotomy for $\Sigma^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $\kappa$: either all $R$-independent sets are of size at most $\kappa$, or there is a $\kappa$-perfect…

For a cardinal kappa and a model M of cardinality kappa let No(M) denote the number of non-isomorphic models of cardinality kappa which are L_{infty,kappa}--equivalent to M. In [Sh:133] Shelah established that when kappa is a weakly compact…

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 prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $\lambda$ for which the sequence $\langle 2^\theta:\theta<\lambda\rangle$ is not eventually constant and the weak diamond fails at $\lambda$. We also…

In this paper, we demonstrate that if, for every $\kappa$-complete fine filter $F$ over $\mathcal{P}_{\kappa}\lambda$, the associated Namba forcing $\mathrm{Nm}(\kappa,\lambda,F)$ is semiproper, then $\square(\mu,{<}\aleph_1)$ fails for all…

Let kappa be an uncountable regular cardinal. Call an equivalence relation on functions from kappa into 2 Sigma_1^1-definable over H(kappa) if there is a first order sentence F and a parameter R subseteq H(kappa) such that functions…

We study higher analogues of the classical independence number on $\omega$. For $\kappa$ regular uncountable, we denote by $i(\kappa)$ the minimal size of a maximal $\kappa$-independent family. We establish ZFC relations between $i(\kappa)$…

Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering.

We show that for any regular cardinal $\kappa$, $\square_{\kappa, 2}$ is consistent with "all $\kappa^+$-Aronszajn trees are special." By a result of Shelah and Stanley this is optimal in the sense that $\square_{\kappa, 2}$ may not be…

Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n<omega the set of alphas with o(alpha)>= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying…

If $X$ is a topological space and $\kappa$ is a cardinal then $\mathsf{BA}_\kappa (X)$ is the statement that for each pair $A, B \subseteq X$ of $\kappa$-dense subsets there is an autohomeomorphism $h:X \to X$ mapping $A$ to $B$. In…

An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC…

We introduce the forcing property of descending distributivity. A forcing $\mathbb{P}$ is $\kappa$-descending distributive if for all decreasing sequences $(D_\alpha)_{\alpha<\kappa}$ of open dense sets, $\bigcap_\alpha D_\alpha$ is open…