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### Keywords

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

# Tag Archives: Aronszajn tree

## The 14th International Workshop on Set Theory in Luminy

I gave an invited talk at the 14th International Workshop on Set Theory in Luminy in Marseille, October 2017. Talk Title: Distributive Aronszajn trees Abstract: It is well-known that that the statement “all $\aleph_1$-Aronszajn trees are special” is consistent with ZFC … Continue reading

Posted in Invited Talks, Squares and Diamonds
Tagged Aronszajn tree, Postprocessing function
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## 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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