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