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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
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
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Mandelbrot set Fodor-type reflection 11P99 tensor product graph PFA(S)[S] Prevalent singular cardinals stationary hitting Shelah's Strong Hypothesis Cardinal function Rainbow sets Aronszajn tree Dushnik-Miller Successor of Singular Cardinal Slim tree Forcing Ostaszewski square xbox Almost countably chromatic weak square Absoluteness incompactness reflection principles Parameterized proxy principle Prikry-type forcing approachability ideal Forcing Axioms coloring number Reduced Power Almost Souslin Uniformization Erdos Cardinal weak diamond Kurepa Hypothesis stationary reflection square Fat stationary set Antichain Coherent tree square principles Uniformly coherent projective Boolean algebra Chromatic number Constructible Universe Generalized Clubs Erdos-Hajnal graphs diamond star Whitehead Problem 05A17 L-space Commutative cancellative semigroups Hindman's Theorem Distributive tree Selective Ultrafilter P-Ideal Dichotomy Poset ccc Chang's conjecture Sakurai's Bell inequality Cardinal Invariants Small forcing OCA Rado's conjecture 20M14 Fast club Rock n' Roll Knaster Jonsson cardinal Axiom R very good scale sap Club Guessing Non-saturation Weakly compact cardinal Square-Brackets Partition Relations Hereditarily Lindelöf space Partition Relations Cohen real Foundations Hedetniemi's conjecture Postprocessing function Minimal Walks Nonspecial tree Singular cardinals combinatorics 05D10 middle diamond Ascent Path Singular coﬁnality Almost-disjoint famiy HOD Souslin Tree b-scale PFA Microscopic Approach Stevo Todorcevic Martin's Axiom Diamond Successor of Regular Cardinal Singular Density Large Cardinals polarized partition relation Universal Sequences free Boolean algebra S-Space

# 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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