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

# 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

## 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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