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Ulam matrix Selective Ultrafilter Vanishing levels sap Universal Sequences PFA(S)[S] Diamond for trees Large Cardinals Singular Density Greatly Mahlo Well-behaved magma Hindman's Theorem Partition Relations square Axiom R Diamond Knaster and friends Chang's conjecture Commutative cancellative semigroups Diamond-sharp Minimal Walks Square-Brackets Partition Relations Generalized descriptive set theory Ramsey theory over partitions approachability ideal Parameterized proxy principle stick stationary hitting indecomposable ultrafilter Closed coloring Was Ulam right Successor of Regular Cardinal Coherent tree Kurepa Hypothesis Cardinal function xbox Absoluteness Uniformly coherent Filter reflection Generalized Clubs Slim tree Martin's Axiom Postprocessing function Ineffable cardinal coloring number Reflecting stationary set incompactness Precaliber weak square Souslin Tree Strong coloring Aronszajn tree ZFC construction Rado's conjecture Strongly compact cardinal C-sequence Non-saturation diamond star Forcing Almost-disjoint family Club Guessing Intersection model Shelah's Strong Hypothesis Almost countably chromatic regressive Souslin tree Open Access Uniformly homogeneous Successor of Singular Cardinal Iterated forcing polarized partition relation Cardinal Invariants Respecting tree Dushnik-Miller Amenable C-sequence higher Baire space Singular cofinality OCA Sakurai's Bell inequality Subadditive Microscopic Approach Ostaszewski square L-space Countryman line SNR nonmeager set Fodor-type reflection Subtle cardinal Mandelbrot set Constructible Universe club_AD Poset Strongly Luzin set Erdos Cardinal Dowker space positive partition relation Hedetniemi's conjecture Antichain free Souslin tree Sierpinski's onto mapping principle strongly bounded groups stationary reflection Rock n' Roll Uniformization Foundations specializable Souslin tree S-Space Knaster Ascent Path countably metacompact transformations Singular cardinals combinatorics Small forcing reflection principles b-scale weak Kurepa tree Distributive tree Fast club Chromatic number Erdos-Hajnal graphs Prevalent singular cardinals Luzin set Almost Souslin Whitehead Problem Sigma-Prikry weak diamond Prikry-type forcing O-space Subtle tree property Nonspecial tree Forcing Axioms PFA Lipschitz reduction AIM forcing ccc Hereditarily Lindelöf space middle diamond very good scale GMA super-Souslin tree tensor product graph Local Club Condensation. 54G20 P-Ideal Dichotomy Weakly compact cardinal unbounded function Fat stationary set Jonsson cardinal Subnormal ideal Analytic sets free Boolean algebra Commutative projection system Rainbow sets Cohen real projective Boolean algebra full tree square principles HOD Reduced Power
Author Archives: Assaf Rinot
Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Joint work with Tanmay Inamdar. Abstract. It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with … Continue reading
Ketonen’s question and other cardinal sins
Joint work with Zhixing You and Jiachen Yuan. Abstract. Intersection models of generic extensions obtained from a commutative projection systems of notions of forcing has recently regained interest, especially in the study of descriptive set theory. Here, we show that … Continue reading
Posted in Compactness, Preprints
Tagged 03E05, 03E35, 03E55, Ascent Path, C-sequence, Commutative projection system, indecomposable ultrafilter, Intersection model, Strongly compact cardinal
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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 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
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
Tagged Jonsson cardinal, Successor of Singular Cardinal
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