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- A strong form of König’s lemma October 21, 2017
- 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
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014

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

# 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

## 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

Posted in Compactness
Tagged approachability ideal, Aronszajn tree, stationary reflection, Weakly compact cardinal
1 Comment

## 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
2 Comments

## 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

## PFA and the tree property at $\aleph_2$

## 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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