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Hereditarily Lindelöf space very good scale Filter reflection Minimal Walks Interval topology on trees Iterated forcing Greatly Mahlo Small forcing Respecting tree ccc Poset Aronszajn tree Forcing Axioms Almost-disjoint family middle diamond Generalized Clubs Rainbow sets Non-saturation transformations O-space Subtle cardinal Diamond-sharp Erdos-Hajnal graphs Fodor-type reflection Selective Ultrafilter coloring number Reflecting stationary set Cohen real projective Boolean algebra regressive Souslin tree Sierpinski's onto mapping principle SNR Mandelbrot set Diamond for trees Almost countably chromatic indecomposable filter strongly bounded groups square principles Precaliber incompactness Cardinal function S-Space Chromatic number Absoluteness Constructible Universe Successor of Regular Cardinal Parameterized proxy principle Subadditive Almost Souslin b-scale Ostaszewski square 54G20 Universal Sequences Partition relations for trees positive partition relation weak diamond HOD Whitehead Problem free Boolean algebra Weakly compact cardinal Well-behaved magma Commutative projection system Ulam matrix Club Guessing Jonsson cardinal reflection principles Closed coloring Martin's Axiom Monotonically far Large Cardinals Postprocessing function Uniformly coherent Hedetniemi's conjecture free Souslin tree Local Club Condensation. Forcing with side conditions Lipschitz reduction Kurepa Hypothesis club_AD Hindman's Theorem Slim tree Strong coloring Nonspecial tree Dushnik-Miller sap Fast club Was Ulam right? PFA Knaster Shelah's Strong Hypothesis Square-Brackets Partition Relations Prikry-type forcing Reduced Power specializable Souslin tree perfectly normal Luzin set square Microscopic Approach Rado's conjecture weak Kurepa tree Foundations countably metacompact stick Dowker space Intersection model Uniformization super-Souslin tree Analytic sets tensor product graph Strongly Luzin set higher Baire space Open Access AIM forcing P-Ideal Dichotomy Ramsey theory over partitions Uniformly homogeneous Sigma-Prikry Ineffable cardinal stationary hitting stationary reflection Chang's conjecture Successor of Singular Cardinal L-space PFA(S)[S] Countryman line Commutative cancellative semigroups Strongly compact cardinal Prevalent singular cardinals weak square Ascent Path full tree Knaster and friends Singular cofinality Entangled linear order Coherent tree Antichain ZFC construction Fat stationary set polarized partition relation Rock n' Roll Subnormal ideal C-sequence approachability ideal diamond star Subtle tree property Generalized descriptive set theory Sakurai's Bell inequality Forcing Souslin Tree Singular Density Ascending path Diamond xbox Partition Relations Amenable C-sequence Singular cardinals combinatorics Vanishing levels Erdos Cardinal unbounded function OCA Cardinal Invariants nonmeager set Distributive tree Axiom R GMA
Author Archives: Assaf Rinot
Set Theory and its Applications in Topology, September 2016
I gave an invited talk at the Set Theory and its Applications in Topology meeting, Oaxaca, September 11-16, 2016. The talk was on the $\aleph_2$-Souslin problem. If you are interested in seeing the effect of a jet lag, the video is … Continue reading
Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
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More notions of forcing add a Souslin tree
Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading
Prikry forcing may add a Souslin tree
A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $\kappa$-Souslin tree? and why is this of interest? My motivation comes from a … Continue reading
The reflection principle $R_2$
A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}[j]\cap\delta\text{ is nonstationary}\}.$$ I wrote there … Continue reading
Posted in Blog
Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
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Ordinal definable subsets of singular cardinals
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading
Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
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Prolific Souslin trees
In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow[\aleph_1]^2_3$ has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found … Continue reading
Posted in Blog, Expository
Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
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A microscopic approach to Souslin-tree constructions. Part I
Joint work with Ari Meir Brodsky. Abstract. We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
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Square with built-in diamond-plus
Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, Respecting tree, square, xbox
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