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

# 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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