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