If κ is an infinite cardinal, a complete Boolean algebra B is called κ-supported if and only if for each sequence 〈bβ : β < κ〉 of elements of B there holds V α<κ W β>α bβ = W A∈[κ]κ V β∈A bβ. We consider the problem of the existence of a non-atomic complete Boolean algebra B which is h2(B)-supported, where h2(B) is the minimal cardinal κ such that B is not (κ, 2)-distributive.

In this note we sketch the proofs of two results in combinatorial set theory. The common theme of the results is singular cardinal combinatorics: they involve the interaction between forcing, large cardinals, PCF theory and versions of Jensen’s weak square principle. 1. Changing cofinality Our first result involves an old question about changes of cofinality. Suppose that V,W are inner models o...

The symbol (XI)κ (with κ ≥ ω) denotes the space XI := Πi∈I Xi with the κ-box topology; this has as base all sets of the form U = Πi∈I Ui with Ui open in Xi and with |{i ∈ I : Ui 6= Xi}| < κ. The symbols w, d and S denote respectively weight, density character, and Souslin number. Generalizing familiar, classical results, the authors show inter alia: Theorem 3.10(b). If κ ≤ α+, |I| = α and each ...

OBJECTIVES The objective was to describe the interobserver agreement between trained chart reviewers and physician reviewers in a multicenter retrospective chart review study of children with cervical spine injuries (CSIs). METHODS Medical records of children younger than 16 years old with cervical spine radiography from 17 Pediatric Emergency Care Applied Research Network (PECARN) hospitals ...

As in the paper, suppose that allies use a threshold strategy according to which ally i supports the ruler if her signal ki is below some threshold k , ki ≤ k , and she rebels if ki > k . Since the signal ki is distributed uniformly on the interval [κ t − ε, κ + ε], a threshold signal k implies the existence of a threshold regime strength κ such that a rebellion fails if κ ≤ κ and succeeds if κ...

Assuming 0 does not exist, κ is an uncountable cardinal and for all cardinals λ with κ ≤ λ < κ+ω, 2 = λ+, we present a “mini-coding” between κ and κ+ω. This allows us to prove that any subset of κ+ω can be coded into a subset, W of κ+ which, further, “reshapes” the interval [κ, κ+), i.e., for all κ < δ < κ+, κ = (card δ)L[W∩δ]. We sketch two applications of this result, assuming 0 does not exis...

For any given uncountable cardinal κ with κ = κ, we present a forcing that is <κ-directed closed, has the κ-cc and introduces a lightface definable well-order of H(κ). We use this to define a global iteration that adds such a well-order for all such κ simultaneously and is capable of preserving the existence of many large cardinals in the universe. 2010 Mathematics Subject Classification: 03E47...

On every set A there is a rigid binary relation i.e. such a relation R ⊆ A × A that there is no homomorphism < A,R >→< A,R > except the identity (Vopěnka et al. [1965]). We prove that for each infinite cardinal number κ if card A ≤ 2 then there exists a relation R ⊆ A×A with the following property ∀x ∈ A ∃ {x}⊆A(x)⊆A cardA(x)≤κ ∀ f :A(x)→A f 6=idA(x) f is not a homomorphism of R which implies t...

Assume G.C.H. and κ is the first uncountable cardinal such that there is a non-free κ-free abelian Whitehead group of cardinality κ. We prove that if all κ-free Abelian group of cardinality κ are Whitehead then κ is necessarily an inaccessible cardinal.