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Vanishing levels Mandelbrot set HOD Well-behaved magma Souslin Tree Knaster countably metacompact Subtle tree property P-Ideal Dichotomy Distributive tree Rock n' Roll Chang's conjecture ccc Coherent tree Uniformly coherent Rado's conjecture PFA(S)[S] Singular Density Open Access Fodor-type reflection weak diamond Cohen real Uniformly homogeneous unbounded function Fat stationary set Generalized descriptive set theory Greatly Mahlo Hindman's Theorem Generalized Clubs stationary hitting Diamond-sharp Ascent Path reflection principles Uniformization Parameterized proxy principle weak Kurepa tree Reduced Power square Forcing Universal Sequences Luzin set Cardinal function stick Small forcing projective Boolean algebra PFA Prikry-type forcing Sakurai's Bell inequality Fast club Large Cardinals Was Ulam right Local Club Condensation. Kurepa Hypothesis Foundations Closed coloring Strongly Luzin set Hedetniemi's conjecture Subnormal ideal positive partition relation Iterated forcing very good scale Axiom R Almost Souslin L-space Non-saturation GMA Singular cofinality Precaliber C-sequence Singular cardinals combinatorics super-Souslin tree polarized partition relation Almost-disjoint family Ulam matrix Analytic sets Ramsey theory over partitions Martin's Axiom indecomposable ultrafilter regressive Souslin tree Dowker space free Boolean algebra Dushnik-Miller incompactness middle diamond ZFC construction Erdos-Hajnal graphs Constructible Universe Prevalent singular cardinals Cardinal Invariants Poset Ostaszewski square Subadditive Absoluteness Commutative cancellative semigroups square principles Amenable C-sequence Aronszajn tree O-space xbox Rainbow sets Sigma-Prikry tensor product graph sap approachability ideal stationary reflection S-Space Forcing Axioms Ineffable cardinal Filter reflection Nonspecial tree Square-Brackets Partition Relations Subtle cardinal Diamond for trees Knaster and friends diamond star Postprocessing function free Souslin tree Selective Ultrafilter Hereditarily Lindelöf space Reflecting stationary set transformations Slim tree Weakly compact cardinal Whitehead Problem Chromatic number Sierpinski's onto mapping principle Almost countably chromatic strongly bounded groups Successor of Singular Cardinal full tree AIM forcing Microscopic Approach Lipschitz reduction Jonsson cardinal Diamond Club Guessing nonmeager set Minimal Walks Successor of Regular Cardinal SNR club_AD 54G20 Antichain Partition Relations Erdos Cardinal b-scale coloring number OCA Shelah's Strong Hypothesis specializable Souslin tree higher Baire space Strong coloring weak square
Tag Archives: Souslin Tree
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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P.O.I. Workshop in pure and descriptive set theory, September 2015
I gave an invited talk at the P.O.I Workshop in pure and descriptive set theory, Torino, September 26, 2015. Title: $\aleph_3$-trees. Abstract: We inspect the constructions of four quite different $\aleph_3$-Souslin trees.
Reduced powers of Souslin trees
Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading
Forcing and its Applications Retrospective Workshop, April 2015
I gave an invited talk at Forcing and its Applications Retrospective Workshop, Toronto, April 1st, 2015. Title: A microscopic approach to Souslin trees constructions Abstract: We present an approach to construct $\kappa$-Souslin trees that is insensitive to the identity of … Continue reading
Posted in Invited Talks
Tagged Microscopic Approach, Parameterized proxy principle, Souslin Tree
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Forcing with a Souslin tree makes $\mathfrak p=\omega_1$
I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading
Forcing with a Souslin tree makes $\mathfrak p=\omega_1$
I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading
The P-Ideal Dichotomy and the Souslin Hypothesis
John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The … Continue reading
Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading
