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- Prikry forcing may add a Souslin tree June 12, 2016
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- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
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### Keywords

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

# 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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