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Fragments of strong compactness, families of partitions and ideal extensions

Laura Fontanella; Pierre Matet — 2016

Fundamenta Mathematicae

We investigate some natural combinatorial principles related to the notion of mild ineffability, and use them to obtain new characterizations of mild ineffable and weakly compact cardinals. We also show that one of these principles may be satisfied by a successor cardinal. Finally, we establish a version for $_{κ} (λ)$ of the canonical Ramsey theorem for pairs.

The tree property at both $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$

Laura Fontanella; Sy David Friedman — 2015

Fundamenta Mathematicae

We force from large cardinals a model of ZFC in which $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model $ℵ_{ω + 2}$ even satisfies the super tree property.

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Fragments of strong compactness, families of partitions and ideal extensions

The tree property at both ℵ ω + 1 and ℵ ω + 2

The tree property at both $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$