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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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- Partitioning the club guessing January 22, 2014

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

# Tag Archives: Aronszajn tree

## 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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