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Fat stationary set HOD stationary hitting club_AD Chang's conjecture Successor of Regular Cardinal Singular cardinals combinatorics Nonspecial tree Large Cardinals Lipschitz reduction square positive partition relation Prevalent singular cardinals Ostaszewski square Generalized Clubs Hedetniemi's conjecture Weakly compact cardinal PFA weak diamond SNR Analytic sets middle diamond Well-behaved magma stick O-space AIM forcing Forcing Axioms Knaster and friends Uniformization Microscopic Approach strongly bounded groups Singular Density higher Baire space Coherent tree free Boolean algebra Erdos-Hajnal graphs Aronszajn tree tensor product graph countably metacompact Sakurai's Bell inequality Diamond-sharp Prikry-type forcing Foundations Hindman's Theorem Almost Souslin Generalized descriptive set theory projective Boolean algebra Universal Sequences P-Ideal Dichotomy Luzin set Ineffable cardinal Postprocessing function square principles coloring number Amenable C-sequence weak Kurepa tree weak square incompactness Reflecting stationary set Subadditive Cardinal Invariants Ascent Path Rock n' Roll Uniformly homogeneous Poset Subnormal ideal Absoluteness Chromatic number Sigma-Prikry Whitehead Problem Fast club Closed coloring diamond star Martin's Axiom very good scale Knaster Strong coloring Commutative cancellative semigroups Square-Brackets Partition Relations Club Guessing Reduced Power Dowker space transformations reflection principles Sierpinski's onto mapping principle Constructible Universe Diamond for trees Non-saturation ccc Ulam matrix polarized partition relation PFA(S)[S] S-Space Small forcing Vanishing levels Partition Relations Almost countably chromatic Precaliber Hereditarily Lindelöf space Almost-disjoint family Iterated forcing Diamond Erdos Cardinal super-Souslin tree Was Ulam right Kurepa Hypothesis Cohen real specializable Souslin tree free Souslin tree b-scale Mandelbrot set sap ZFC construction OCA Fodor-type reflection xbox unbounded function Shelah's Strong Hypothesis Open Access Rado's conjecture Ramsey theory over partitions Rainbow sets Greatly Mahlo stationary reflection C-sequence Parameterized proxy principle Successor of Singular Cardinal L-space full tree regressive Souslin tree Jonsson cardinal Local Club Condensation. Selective Ultrafilter Souslin Tree Singular cofinality Uniformly coherent approachability ideal Minimal Walks nonmeager set Subtle cardinal Filter reflection Strongly Luzin set Axiom R Distributive tree 54G20 GMA Antichain Subtle tree property Cardinal function Dushnik-Miller Forcing indecomposable ultrafilter Slim tree
Author Archives: Assaf Rinot
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 only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by … Continue reading
Posted in Preprints, Squares and Diamonds
Tagged Diamond, Diamond for trees, Iterated forcing, Kurepa Hypothesis, weak Kurepa tree
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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
Posted in Preprints, Souslin Hypothesis
Tagged C-sequence, free Souslin tree, Parameterized proxy principle, Souslin Tree, specializable Souslin tree, square principles, xbox
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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
Posted in Compactness, Preprints
Tagged Forcing Axioms, indecomposable ultrafilter, Subadditive, unbounded function
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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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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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The vanishing levels of a tree
Joint work with Shira Yadai and Zhixing You. Abstract. We initiate the study of the spectrum $Vspec(\kappa)$ of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa$-tree $\mathbf T$. The latter is an invariant in … Continue reading
Posted in Preprints, Souslin Hypothesis
Tagged Almost-disjoint family, C-sequence, Coherent tree, Parameterized proxy principle, regressive Souslin tree, Subtle tree property, Uniformly homogeneous, Vanishing levels
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Full Souslin trees at small cardinals
Joint work with Shira Yadai and Zhixing You. Abstract. A $\kappa$-tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $\kappa$-Souslin tree may consistently exist. Shelah gave an affirmative … Continue reading
Posted in Preprints, Souslin Hypothesis
Tagged Diamond for trees, full tree, Greatly Mahlo, Microscopic Approach, Parameterized proxy principle, Subtle cardinal, tensor product graph
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A counterexample related to a theorem of Komjáth and Weiss
Joint work with Rodrigo Rey Carvalho. Abstract. In a paper from 1987, Komjath and Weiss proved that for every regular topological space $X$ of character less than $\mathfrak b$, if $X\rightarrow(\text{top }{\omega+1})^1_\omega$, then $X\rightarrow(\text{top }{\alpha})^1_\omega$ for all $\alpha<\omega_1$. In addition, … Continue reading
Posted in Partition Relations, Preprints, Topology
Tagged 03E02, 54G20, ZFC construction
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A Shelah group in ZFC
Joint work with Márk Poór. Abstract. In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group $G$ that moreover admits an integer $n$ satisfying … Continue reading
Posted in Groups, Preprints
Tagged 03E02, 03E75, 20A15, 20E15, 20F06, Jonsson cardinal, Strong coloring, strongly bounded groups, Subadditive, ZFC construction
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