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- Prikry forcing may add a Souslin tree June 12, 2016
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- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
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Selective Ultrafilter Small forcing Axiom R Large Cardinals Singular coﬁnality Successor of Regular Cardinal Fodor-type reflection Cohen real Uniformly coherent P-Ideal Dichotomy xbox 20M14 11P99 OCA Stevo Todorcevic Commutative cancellative semigroups Slim tree Distributive tree Dushnik-Miller polarized partition relation PFA Coherent tree Minimal Walks Almost Souslin Forcing Chang's conjecture PFA(S)[S] middle diamond Singular Density Ostaszewski square Constructible Universe Sakurai's Bell inequality ccc Hindman's Theorem tensor product graph b-scale Forcing Axioms diamond star Aronszajn tree Rainbow sets Singular cardinals combinatorics Poset stationary hitting Souslin Tree Ascent Path HOD coloring number incompactness Square-Brackets Partition Relations Diamond weak square Nonspecial tree Hereditarily Lindelöf space Almost countably chromatic Shelah's Strong Hypothesis Absoluteness Fast club Reduced Power very good scale Chromatic number Partition Relations Weakly compact cardinal Knaster square principles L-space weak diamond Erdos-Hajnal graphs Kurepa Hypothesis S-Space approachability ideal sap Rado's conjecture super-Souslin tree Martin's Axiom Jonsson cardinal Non-saturation Cardinal Invariants Mandelbrot set Hedetniemi's conjecture Generalized Clubs Parameterized proxy principle square Erdos Cardinal Postprocessing function Fat stationary set stationary reflection projective Boolean algebra Microscopic Approach Universal Sequences Whitehead Problem Club Guessing Antichain Luzin set Foundations Cardinal function free Boolean algebra reflection principles Prevalent singular cardinals Uniformization Almost-disjoint famiy Rock n' Roll Prikry-type forcing 05A17 Successor of Singular Cardinal

# Tag Archives: Aronszajn tree

## Distributive Aronszajn trees

Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … Continue reading

## The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading

## Chain conditions of products, and weakly compact cardinals

Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading

Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
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## PFA and the tree property at $\aleph_2$

Recall that a poset $\langle T,\le\rangle$ is said to be a $\lambda^+$-Aronszajn tree, if it isomorphic to a poset $(\mathcal T,\subseteq)$ of the form: $\emptyset\in \mathcal T\subseteq{}^{<\lambda^+}\lambda$; Write $\mathcal T_\alpha:=\{\sigma\in\mathcal T\mid \text{dom}(\sigma)=\alpha\}$; for all $\alpha<\lambda^+$, $\mathcal T_\alpha$ has size $\le\lambda$, … Continue reading

## A cofinality-preserving small forcing may introduce a special Aronszajn tree

Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
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