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