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Dushnik-Miller Almost Souslin Intersection model HOD Precaliber Aronszajn tree Generalized Clubs O-space Minimal Walks Interval topology on trees Subnormal ideal Greatly Mahlo Generalized descriptive set theory Antichain Fat stationary set weak square Kurepa Hypothesis Weakly compact cardinal L-space projective Boolean algebra Sierpinski's onto mapping principle Distributive tree Whitehead Problem Non-saturation Nonspecial tree Cohen real sap Was Ulam right? GMA Analytic sets Chromatic number Rado's conjecture Cardinal function Forcing with side conditions Successor of Singular Cardinal square full tree Almost-disjoint family Postprocessing function Diamond-sharp weak diamond Ostaszewski square perfectly normal Poset Selective Ultrafilter Ascent Path P-Ideal Dichotomy Iterated forcing transformations OCA Ineffable cardinal free Boolean algebra ZFC construction Parameterized proxy principle 54G20 Subtle cardinal Entangled linear order Local Club Condensation. Erdos-Hajnal graphs Mandelbrot set ccc Singular cofinality AIM forcing Hereditarily Lindelöf space Monotonically far Cardinal Invariants unbounded function Amenable C-sequence Diamond for trees weak Kurepa tree Foundations Dowker space Luzin set Lipschitz reduction Well-behaved magma Jonsson cardinal regressive Souslin tree Ramsey theory over partitions Sakurai's Bell inequality Constructible Universe Uniformization S-Space Subadditive Countryman line xbox Fast club higher Baire space coloring number Club Guessing Diamond SNR Small forcing Closed coloring Uniformly coherent Souslin Tree Reduced Power stick Microscopic Approach polarized partition relation Shelah's Strong Hypothesis Filter reflection C-sequence strongly bounded groups Reflecting stationary set Axiom R Rock n' Roll Singular Density Sigma-Prikry Knaster Successor of Regular Cardinal Subtle tree property Ulam matrix Knaster and friends Coherent tree Chang's conjecture Square-Brackets Partition Relations stationary reflection club_AD Martin's Axiom Large Cardinals Ascending path Commutative projection system Partition Relations b-scale very good scale super-Souslin tree PFA(S)[S] Commutative cancellative semigroups indecomposable filter nonmeager set Singular cardinals combinatorics square principles Strong coloring Erdos Cardinal tensor product graph Universal Sequences Rainbow sets diamond star countably metacompact Partition relations for trees Respecting tree Absoluteness positive partition relation middle diamond Prikry-type forcing Forcing Axioms Forcing specializable Souslin tree Strongly compact cardinal stationary hitting free Souslin tree Open Access Strongly Luzin set Uniformly homogeneous reflection principles Prevalent singular cardinals incompactness Almost countably chromatic Fodor-type reflection approachability ideal Vanishing levels Slim tree PFA Hindman's Theorem Hedetniemi's conjecture
Tag Archives: Aronszajn tree
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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