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- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Genearlizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

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Universal Sequences OCA 05D10 HOD Ostaszewski square Coherent tree Forcing Axioms Almost Souslin stationary hitting approachability ideal Antichain 05A17 diamond star xbox Forcing projective Boolean algebra ccc Reduced Power Uniformization Absoluteness middle diamond Generalized Clubs Rado's conjecture square Erdos Cardinal Cardinal function Parameterized proxy principle Hindman's Theorem Diamond Souslin Tree Non-saturation Singular Cofinality Jonsson cardinal Hedetniemi's conjecture Microscopic Approach Partition Relations Almost countably chromatic Kurepa Hypothesis Singular Density Commutative cancellative semigroups Poset Fat stationary set Rainbow sets Fodor-type reflection Foundations Singular cardinals combinatorics Almost-disjoint famiy coloring number 11P99 stationary reflection Mandelbrot set L-space Sakurai's Bell inequality very good scale reflection principles Axiom R weak diamond Large Cardinals P-Ideal Dichotomy Shelah's Strong Hypothesis Knaster Successor of Regular Cardinal tensor product graph Weakly compact cardinal free Boolean algebra sap PFA Slim tree Constructible Universe weak square 20M14 Erdos-Hajnal graphs Cohen real Square-Brackets Partition Relations Cardinal Invariants Chang's conjecture Singular coﬁnality Successor of Singular Cardinal Aronszajn tree Rock n' Roll Dushnik-Miller Fast club Martin's Axiom Small forcing b-scale PFA(S)[S] S-Space Ascent Path incompactness Minimal Walks Prevalent singular cardinals Whitehead Problem Prikry-type forcing Stevo Todorcevic Selective Ultrafilter Hereditarily Lindelöf space Chromatic number Club Guessing polarized partition relation

# Tag Archives: Aronszajn tree

## 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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