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Subnormal ideal Selective Ultrafilter weak square GMA Jonsson cardinal Microscopic Approach Dowker space stationary reflection countably metacompact Greatly Mahlo higher Baire space Singular cofinality polarized partition relation free Boolean algebra reflection principles weak diamond coloring number sap Almost Souslin Subtle tree property xbox Local Club Condensation. indecomposable ultrafilter Uniformly coherent strongly bounded groups Was Ulam right Non-saturation AIM forcing 54G20 S-Space stationary hitting Successor of Singular Cardinal Coherent tree Diamond stick Rado's conjecture Precaliber positive partition relation Rainbow sets Partition Relations Analytic sets Reflecting stationary set Subadditive Luzin set Souslin Tree C-sequence free Souslin tree Large Cardinals b-scale incompactness Kurepa Hypothesis Hindman's Theorem unbounded function L-space Martin's Axiom Generalized descriptive set theory Weakly compact cardinal Ramsey theory over partitions Parameterized proxy principle specializable Souslin tree Square-Brackets Partition Relations Prevalent singular cardinals Knaster and friends middle diamond Sigma-Prikry Lipschitz reduction Diamond for trees Prikry-type forcing Cardinal Invariants approachability ideal diamond star Uniformly homogeneous Singular Density square Knaster Closed coloring transformations Commutative cancellative semigroups Minimal Walks Shelah's Strong Hypothesis P-Ideal Dichotomy Open Access nonmeager set Cohen real Chromatic number Erdos Cardinal Axiom R Rock n' Roll Hereditarily Lindelöf space Ineffable cardinal Foundations tensor product graph Forcing Absoluteness Hedetniemi's conjecture Cardinal function Small forcing Sakurai's Bell inequality Uniformization Aronszajn tree Successor of Regular Cardinal Amenable C-sequence Fast club Almost-disjoint family Reduced Power Dushnik-Miller Ulam matrix Whitehead Problem Fodor-type reflection Fat stationary set Antichain Diamond-sharp Filter reflection SNR projective Boolean algebra regressive Souslin tree Universal Sequences club_AD PFA(S)[S] Sierpinski's onto mapping principle Well-behaved magma Postprocessing function PFA Subtle cardinal Chang's conjecture Poset very good scale ccc Iterated forcing Erdos-Hajnal graphs full tree Constructible Universe OCA super-Souslin tree Ostaszewski square square principles ZFC construction Ascent Path O-space HOD Club Guessing Singular cardinals combinatorics Forcing Axioms Nonspecial tree Vanishing levels Almost countably chromatic Mandelbrot set Generalized Clubs Strong coloring Slim tree Distributive tree Strongly Luzin set
Tag Archives: Successor of Singular Cardinal
Perspectives on Set Theory, November 2023
I gave an invited talk at the Perspectives on Set Theory conference, November 2023. Talk Title: May the successor of a singular cardinal be Jónsson? Abstract: We’ll survey what’s known about the question in the title and collect ten open … Continue reading
Posted in Invited Talks, Open Problems, Partition Relations
Tagged Jonsson cardinal, Successor of Singular 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
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
Putting a diamond inside the square
Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
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A cofinality-preserving small forcing may introduce a special Aronszajn tree
Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
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The failure of diamond on a reflecting stationary set
Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading
A relative of the approachability ideal, diamond and non-saturation
Abstract: Let $\lambda$ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often. In this paper, for a subset $S\subset\lambda^+$, a normal subideal of … Continue reading
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading