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