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