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Singular Density Diamond Cardinal function Kurepa Hypothesis ZFC construction Diamond-sharp perfectly normal Erdos Cardinal transformations Uniformly homogeneous Antichain Commutative cancellative semigroups Sigma-Prikry Universal Sequences Weakly compact cardinal weak Kurepa tree coloring number incompactness Successor of Regular Cardinal full tree Subadditive Souslin Tree higher Baire space Rainbow sets Subnormal ideal Reflecting stationary set Dowker space stationary reflection Intersection model Ramsey theory over partitions Subtle cardinal Was Ulam right? Partition Relations indecomposable filter Generalized descriptive set theory Filter reflection Ulam matrix Well-behaved magma Minimal Walks Generalized Clubs Whitehead Problem Amenable C-sequence C-sequence Analytic sets Strongly compact cardinal Cohen real Constructible Universe club_AD L-space 54G20 square Knaster Erdos-Hajnal graphs unbounded function Luzin set Precaliber Entangled linear order Uniformization Monotonically far countably metacompact S-Space diamond star very good scale super-Souslin tree HOD Local Club Condensation. Forcing Axioms Shelah's Strong Hypothesis polarized partition relation Large Cardinals O-space Hedetniemi's conjecture Ascending path Forcing ccc Iterated forcing Small forcing Commutative projection system Open Access Martin's Axiom Slim tree Hereditarily Lindelöf space PFA Hindman's Theorem b-scale Strongly Luzin set GMA Sakurai's Bell inequality free Boolean algebra Cardinal Invariants Distributive tree Almost Souslin Almost-disjoint family Respecting tree middle diamond Axiom R P-Ideal Dichotomy Countryman line Diamond for trees Closed coloring Lipschitz reduction OCA projective Boolean algebra stick specializable Souslin tree Strong coloring Absoluteness reflection principles Jonsson cardinal Microscopic Approach Ascent Path Knaster and friends Selective Ultrafilter Chromatic number Aronszajn tree Prikry-type forcing Ineffable cardinal Forcing with side conditions square principles Reduced Power tensor product graph Coherent tree Fast club Almost countably chromatic Singular cardinals combinatorics Subtle tree property SNR Rado's conjecture xbox Dushnik-Miller Sierpinski's onto mapping principle Vanishing levels nonmeager set Fat stationary set Interval topology on trees Prevalent singular cardinals Chang's conjecture Foundations Square-Brackets Partition Relations Non-saturation AIM forcing Fodor-type reflection Parameterized proxy principle weak square Rock n' Roll strongly bounded groups Poset regressive Souslin tree Mandelbrot set Partition relations for trees Uniformly coherent approachability ideal Greatly Mahlo Nonspecial tree Ostaszewski square Singular cofinality free Souslin tree Club Guessing PFA(S)[S] sap stationary hitting Successor of Singular Cardinal Postprocessing function weak diamond positive partition relation
Author Archives: Assaf Rinot
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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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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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
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, Open Access, Prikry-type forcing, 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, Publications
Tagged 03E02, 03E75, 20A15, 20E15, 20F06, Jonsson cardinal, Open Access, Strong coloring, strongly bounded groups, Subadditive, ZFC construction
3 Comments
Gdańsk Logic Conference, May 2023
I gave an invited talk at the first Gdańsk Logic Conference, May 2023. Talk Title: Was Ulam right? Abstract: An Ulam matrix is one of the earliest gems of infinite combinatorics. Around the same time of its discovery, another Polish … Continue reading
A series of lectures on Club_AD, February–March 2023
As part of the Thematic Program on Set Theoretic Methods in Algebra, Dynamics and Geometry (Fields Institute, January–June, 2023), Spencer Unger and I delivered a Graduate Course on Set Theory, Algebra and Analysis. My part of the course was a … Continue reading
Winter School in Abstract Analysis, January 2023
I gave a 3-lecture tutorial at the Winter School in Abstract Analysis in Steken, January 2023. Title: Club guessing Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove … Continue reading
Sums of triples in Abelian groups
Joint work with Ido Feldman. Abstract. Motivated by a problem in additive Ramsey theory, we extend Todorcevic’s partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for … Continue reading
