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