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