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Fat stationary set Commutative cancellative semigroups Iterated forcing Uniformization Rock n' Roll Amenable C-sequence Greatly Mahlo Shelah's Strong Hypothesis Slim tree AIM forcing O-space Almost-disjoint family Antichain Large Cardinals strongly bounded groups regressive Souslin tree Singular cardinals combinatorics Successor of Singular Cardinal Closed coloring Respecting tree countably metacompact Forcing Axioms OCA Successor of Regular Cardinal diamond star higher Baire space Chang's conjecture square Chromatic number Ostaszewski square Partition Relations Singular Density Axiom R coloring number tensor product graph Weakly compact cardinal Distributive tree Whitehead Problem free Souslin tree 54G20 Subnormal ideal stick Fodor-type reflection Reflecting stationary set Strong coloring transformations reflection principles Was Ulam right? C-sequence Uniformly homogeneous Square-Brackets Partition Relations ccc positive partition relation S-Space Luzin set Uniformly coherent Jonsson cardinal Aronszajn tree polarized partition relation incompactness Hereditarily Lindelöf space Sakurai's Bell inequality Diamond-sharp Ramsey theory over partitions weak Kurepa tree Fast club Erdos Cardinal nonmeager set Intersection model ZFC construction Foundations Cardinal function very good scale Almost countably chromatic Analytic sets Vanishing levels super-Souslin tree Sigma-Prikry PFA(S)[S] Lipschitz reduction Poset approachability ideal Subtle tree property Rado's conjecture xbox Club Guessing Universal Sequences Coherent tree Postprocessing function Knaster Strongly Luzin set Almost Souslin Singular cofinality Hindman's Theorem Ulam matrix L-space Parameterized proxy principle PFA Local Club Condensation. Filter reflection Constructible Universe Prikry-type forcing Hedetniemi's conjecture Absoluteness Souslin Tree Generalized descriptive set theory Diamond for trees P-Ideal Dichotomy Non-saturation Commutative projection system Ascent Path Diamond Sierpinski's onto mapping principle Martin's Axiom Prevalent singular cardinals HOD Erdos-Hajnal graphs Mandelbrot set Cohen real Dowker space unbounded function stationary hitting Knaster and friends stationary reflection club_AD middle diamond Microscopic Approach SNR Selective Ultrafilter Ineffable cardinal sap Countryman line Well-behaved magma b-scale Open Access square principles free Boolean algebra GMA Kurepa Hypothesis specializable Souslin tree weak square Subadditive weak diamond Precaliber Nonspecial tree Minimal Walks Generalized Clubs indecomposable ultrafilter Reduced Power Strongly compact cardinal Dushnik-Miller Subtle cardinal Forcing Cardinal Invariants full tree Rainbow sets Small forcing projective Boolean algebra
Author Archives: Assaf Rinot
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
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, 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, Preprints
Tagged 03E02, 03E75, 20A15, 20E15, 20F06, Jonsson cardinal, Strong coloring, strongly bounded groups, Subadditive, ZFC construction
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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
Posted in Invited Talks
Tagged Ulam matrix, Was Ulam right?
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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
A club guessing toolbox I
Joint work with Tanmay Inamdar. Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have … Continue reading
Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems
Joint work with Menachem Kojman and Juris Steprāns. Abstract. In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in … Continue reading
