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Large Cardinals Chang's conjecture Uniformly homogeneous Sigma-Prikry Sakurai's Bell inequality Interval topology on trees ZFC construction Commutative cancellative semigroups Knaster Fast club PFA(S)[S] Nonspecial tree higher Baire space Forcing with side conditions full tree Luzin set Singular Density Hedetniemi's conjecture Aronszajn tree Universal Sequences Amenable C-sequence Ramsey theory over partitions square Reduced Power Partition Relations middle diamond Countryman line Souslin Tree Diamond-sharp Diamond weak Kurepa tree Fat stationary set Axiom R countably metacompact Prikry-type forcing 54G20 Erdos-Hajnal graphs Absoluteness Generalized descriptive set theory Martin's Axiom positive partition relation Distributive tree b-scale Selective Ultrafilter Greatly Mahlo Foundations Shelah's Strong Hypothesis transformations AIM forcing Uniformly coherent Subtle cardinal stick Jonsson cardinal Singular cardinals combinatorics perfectly normal Postprocessing function P-Ideal Dichotomy Respecting tree Ascent Path Cardinal function Ulam matrix weak diamond Almost-disjoint family Forcing unbounded function Cardinal Invariants Entangled linear order Kurepa Hypothesis Prevalent singular cardinals Iterated forcing stationary hitting free Boolean algebra super-Souslin tree Precaliber Whitehead Problem Hindman's Theorem Hereditarily Lindelöf space Strongly compact cardinal Minimal Walks S-Space weak square Dowker space regressive Souslin tree Cohen real strongly bounded groups Diamond for trees Microscopic Approach Chromatic number GMA Small forcing Square-Brackets Partition Relations Open Access Knaster and friends Forcing Axioms very good scale Monotonically far Subadditive stationary reflection Poset diamond star Rado's conjecture projective Boolean algebra Intersection model Non-saturation reflection principles Ineffable cardinal C-sequence Strongly Luzin set Almost countably chromatic Successor of Singular Cardinal PFA Mandelbrot set Coherent tree Almost Souslin Rainbow sets approachability ideal xbox Sierpinski's onto mapping principle Dushnik-Miller Antichain free Souslin tree Local Club Condensation. Was Ulam right? Well-behaved magma HOD OCA Vanishing levels Uniformization Slim tree L-space Ostaszewski square square principles Commutative projection system Generalized Clubs Fodor-type reflection Club Guessing Singular cofinality polarized partition relation Analytic sets Weakly compact cardinal Strong coloring specializable Souslin tree indecomposable filter nonmeager set Reflecting stationary set SNR Partition relations for trees Closed coloring Rock n' Roll Erdos Cardinal Lipschitz reduction ccc tensor product graph Successor of Regular Cardinal club_AD Filter reflection Constructible Universe incompactness Ascending path Subnormal ideal O-space Subtle tree property sap Parameterized proxy principle coloring number
Category Archives: Singular Cardinals Combinatorics
May the successor of a singular cardinal be Jonsson?
Abstract: Whether the successor of a singular cardinal can be Jónsson is a very old and famous open problem in set theory. Here, we collect necessary conditions for an affirmative answer, and put forward a list of closely-related questions in … Continue reading
Posted in Open Problems, Singular Cardinals Combinatorics
Tagged Jonsson cardinal, Open Access
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Sigma-Prikry forcing III: Down to Aleph_omega
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classical results of … Continue reading
Sigma-Prikry forcing II: Iteration Scheme
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\Sigma$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers … Continue reading
Knaster and friends II: The C-sequence number
Joint work with Chris Lambie-Hanson. Abstract. Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable … Continue reading
Sigma-Prikry forcing I: The Axioms
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality … Continue reading
Partitioning a reflecting stationary set
Joint work with Maxwell Levine. Abstract. We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer … Continue reading
Ordinal definable subsets of singular cardinals
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading
Aspects of singular cofinality
Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer … Continue reading
On the consistency strength of the Milner-Sauer conjecture
Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading
Antichains in partially ordered sets of singular cofinality
Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The main result of of this … Continue reading
Posted in Publications, Singular Cardinals Combinatorics
Tagged 03E04, 03E35, 06A07, Antichain, Poset, Singular cofinality
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