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