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