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stationary reflection Ramsey theory over partitions Open Access unbounded function higher Baire space free Souslin tree Diamond-sharp Strongly compact cardinal Almost countably chromatic Ineffable cardinal Constructible Universe coloring number Local Club Condensation. Was Ulam right? Sakurai's Bell inequality P-Ideal Dichotomy Entangled linear order O-space Nonspecial tree Cohen real Knaster strongly bounded groups square incompactness xbox tensor product graph Ostaszewski square Countryman line Subtle tree property Filter reflection HOD Kurepa Hypothesis Reflecting stationary set transformations Large Cardinals Forcing Axioms Sigma-Prikry Almost-disjoint family Fat stationary set Axiom R positive partition relation Erdos Cardinal 54G20 Foundations Rock n' Roll weak square Aronszajn tree Forcing Diamond for trees Precaliber Microscopic Approach Commutative cancellative semigroups Ascending path L-space PFA AIM forcing Fast club Hindman's Theorem Diamond Sierpinski's onto mapping principle ccc Minimal Walks Closed coloring Almost Souslin Partition Relations Hedetniemi's conjecture Singular Density PFA(S)[S] Slim tree Absoluteness countably metacompact Iterated forcing GMA Reduced Power middle diamond Parameterized proxy principle Subtle cardinal Dowker space S-Space Hereditarily Lindelöf space Square-Brackets Partition Relations full tree stick Commutative projection system Successor of Singular Cardinal Analytic sets Ulam matrix weak diamond Well-behaved magma Cardinal function very good scale approachability ideal Amenable C-sequence Monotonically far Martin's Axiom Jonsson cardinal Uniformization Whitehead Problem Greatly Mahlo Generalized descriptive set theory Successor of Regular Cardinal Selective Ultrafilter Respecting tree square principles Forcing with side conditions Postprocessing function Ascent Path Coherent tree indecomposable filter Lipschitz reduction Luzin set Strong coloring Strongly Luzin set free Boolean algebra Uniformly coherent Weakly compact cardinal Mandelbrot set b-scale nonmeager set Dushnik-Miller club_AD Small forcing Intersection model Generalized Clubs Knaster and friends Antichain Rado's conjecture projective Boolean algebra sap Partition relations for trees Subnormal ideal polarized partition relation Interval topology on trees Poset Non-saturation regressive Souslin tree diamond star Souslin Tree Club Guessing ZFC construction Chang's conjecture stationary hitting super-Souslin tree Fodor-type reflection Singular cardinals combinatorics reflection principles Distributive tree Rainbow sets weak Kurepa tree Uniformly homogeneous OCA Subadditive C-sequence perfectly normal specializable Souslin tree Prikry-type forcing Cardinal Invariants Erdos-Hajnal graphs Vanishing levels Singular cofinality SNR Shelah's Strong Hypothesis Universal Sequences Prevalent singular cardinals Chromatic number
Author Archives: Assaf Rinot
Ketonen’s question and other cardinal sins
Joint work with Zhixing You and Jiachen Yuan. Abstract. Answering a question of Ketonen from the late 1970’s, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing … Continue reading
120 Years of Choice, July 2024
I gave an invited talk at the 120 Years of Choice conference, July 2024. Talk Title: Mathematician’s best friend Abstract: Jensen’s diamond is a very useful postulate. It is well-known that it implies the continuum hypothesis, that it is equivalent … Continue reading
Diamond on Kurepa trees
Joint work with Ziemek Kostana and Saharon Shelah. Abstract. We introduce a new weak variation of diamond that is meant to guess only the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by … Continue reading
Proxy principles in combinatorial set theory
Joint work with Ari Meir Brodsky and Shira Yadai. Abstract. The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper as new foundations for the construction of $\kappa$-Souslin trees in a uniform way that does not … Continue reading
Squares, ultrafilters and forcing axioms
Joint work with Chris Lambie-Hanson and Jing Zhang. Abstract. We study the interplay of the three families of combinatorial objects or principles. Specifically, we show the following. Strong forcing axioms, in general incompatible with the existence of indexed squares, can … Continue reading
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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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
Tagged Jonsson cardinal, Successor of Singular Cardinal
Comments Off on Perspectives on Set Theory, November 2023
The vanishing levels of a tree
Joint work with Shira Yadai and Zhixing You. Abstract. We initiate the study of the spectrum of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa$-tree $\mathbf T$. This is an invariant in the … Continue reading
Posted in Preprints, Souslin Hypothesis
Tagged Almost-disjoint family, Ascent Path, C-sequence, Coherent tree, Dowker space, Open Access, Parameterized proxy principle, regressive Souslin tree, Respecting tree, Subtle tree property, Uniformly homogeneous, Vanishing levels, weak Kurepa tree
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