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