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