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Respecting tree incompactness Cardinal function Analytic sets super-Souslin tree Luzin set free Souslin tree weak diamond Martin's Axiom weak square Axiom R Fat stationary set Intersection model Selective Ultrafilter C-sequence Antichain Subnormal ideal perfectly normal Iterated forcing countably metacompact Singular cofinality polarized partition relation square Almost Souslin Parameterized proxy principle Local Club Condensation. weak Kurepa tree Strongly Luzin set Uniformly homogeneous Open Access Strongly compact cardinal Microscopic Approach AIM forcing Coherent tree Fast club Aronszajn tree ZFC construction Cohen real O-space Chromatic number Countryman line Entangled linear order reflection principles stick Commutative cancellative semigroups Amenable C-sequence Jonsson cardinal transformations PFA(S)[S] Almost-disjoint family Weakly compact cardinal Prikry-type forcing Rainbow sets coloring number L-space Forcing with side conditions Successor of Regular Cardinal unbounded function Singular Density Whitehead Problem Sakurai's Bell inequality Small forcing Large Cardinals Reflecting stationary set Fodor-type reflection HOD Generalized Clubs Club Guessing Dowker space Almost countably chromatic Uniformly coherent Monotonically far Lipschitz reduction strongly bounded groups Successor of Singular Cardinal Subadditive Distributive tree nonmeager set ccc xbox Subtle cardinal Diamond Cardinal Invariants PFA indecomposable filter Rado's conjecture b-scale Rock n' Roll GMA Universal Sequences middle diamond Ostaszewski square Knaster Absoluteness Strong coloring Mandelbrot set projective Boolean algebra S-Space 54G20 square principles very good scale higher Baire space Partition relations for trees Partition Relations Reduced Power club_AD Generalized descriptive set theory Ineffable cardinal stationary reflection Shelah's Strong Hypothesis Closed coloring free Boolean algebra Sigma-Prikry Forcing Axioms OCA regressive Souslin tree Non-saturation Erdos Cardinal Diamond for trees Well-behaved magma Knaster and friends SNR Singular cardinals combinatorics Diamond-sharp Minimal Walks positive partition relation Souslin Tree Poset Sierpinski's onto mapping principle Foundations Kurepa Hypothesis Dushnik-Miller P-Ideal Dichotomy Postprocessing function Greatly Mahlo Hereditarily Lindelöf space Filter reflection diamond star Hindman's Theorem full tree Hedetniemi's conjecture Ascending path Constructible Universe Ramsey theory over partitions Ascent Path Chang's conjecture Erdos-Hajnal graphs Precaliber stationary hitting approachability ideal Uniformization Vanishing levels tensor product graph sap Forcing Commutative projection system Nonspecial tree Was Ulam right? Ulam matrix Slim tree Subtle tree property Prevalent singular cardinals specializable Souslin tree Interval topology on trees Square-Brackets Partition Relations
Tag Archives: Aronszajn tree
Partition relations for trees II: Entangled linear orders
Joint work with Tanmay Inamdar. Abstract. We find new sufficient conditions for the existence of entangled linear orders. The constructions use anti-Ramsey colourings for trees, and they provide a fine control on the extent of entangledness. As an application, from … Continue reading
Posted in Partition Relations, Preprints
Tagged Aronszajn tree, Entangled linear order, Partition relations for trees, Strongly Luzin set
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Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Joint work with Tanmay Inamdar. Abstract. We investigate global structural properties of linear orders of a fixed infinite size. It is classical that the countable linear orders and the continuum-sized orders exhibit contrasting behaviours. Modern results show that strong extensions … Continue reading
Posted in Basis problems, Partition Relations, Preprints
Tagged Aronszajn tree, Ascending path, Club Guessing, Countryman line, Entangled linear order, Minimal Walks, Monotonically far, Partition relations for trees, Strong coloring, Subtle tree property, Vanishing levels, ZFC construction
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11th Young Set Theory Workshop, June 2018
I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that … Continue reading
Posted in Invited Talks
Tagged Aronszajn tree, C-sequence, incompactness, Knaster, Minimal Walks, Postprocessing function, square
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The 14th International Workshop on Set Theory in Luminy, October 2017
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
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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