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