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