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