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