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Aronszajn tree Cardinal function full tree strongly bounded groups S-Space Well-behaved magma projective Boolean algebra Cardinal Invariants weak square Jonsson cardinal free Boolean algebra higher Baire space Ramsey theory over partitions Monotonically far diamond star Universal Sequences Interval topology on trees very good scale countably metacompact square Kurepa Hypothesis Hedetniemi's conjecture Slim tree Hindman's Theorem Closed coloring Whitehead Problem Sigma-Prikry unbounded function Chromatic number specializable Souslin tree Entangled linear order Fodor-type reflection Square-Brackets Partition Relations nonmeager set Knaster HOD Singular cofinality C-sequence Diamond PFA Club Guessing Sakurai's Bell inequality Foundations Ostaszewski square Antichain free Souslin tree Subnormal ideal tensor product graph coloring number Postprocessing function stationary hitting Iterated forcing Fat stationary set Cohen real Large Cardinals Forcing with side conditions Respecting tree middle diamond reflection principles Rainbow sets b-scale Intersection model Singular cardinals combinatorics Successor of Regular Cardinal Coherent tree incompactness Almost-disjoint family Weakly compact cardinal weak diamond Souslin Tree Axiom R Ascent Path Small forcing AIM forcing Luzin set Strongly Luzin set Ascending path Greatly Mahlo Subtle cardinal L-space Fast club xbox Uniformly homogeneous Prevalent singular cardinals Sierpinski's onto mapping principle Commutative cancellative semigroups Strong coloring Ulam matrix weak Kurepa tree Precaliber Filter reflection Amenable C-sequence Reflecting stationary set Parameterized proxy principle Singular Density sap Local Club Condensation. Was Ulam right? OCA Subtle tree property regressive Souslin tree Poset Hereditarily Lindelöf space approachability ideal Martin's Axiom Subadditive Almost Souslin ZFC construction Non-saturation Open Access Diamond for trees Rado's conjecture Uniformly coherent P-Ideal Dichotomy Chang's conjecture Commutative projection system GMA Generalized descriptive set theory Countryman line stationary reflection Minimal Walks super-Souslin tree O-space Analytic sets Absoluteness transformations Selective Ultrafilter Distributive tree Generalized Clubs Constructible Universe Knaster and friends Vanishing levels square principles Rock n' Roll Dushnik-Miller Successor of Singular Cardinal Shelah's Strong Hypothesis SNR Diamond-sharp Forcing Axioms Prikry-type forcing Dowker space PFA(S)[S] Uniformization perfectly normal Nonspecial tree Reduced Power indecomposable filter club_AD Erdos Cardinal 54G20 Forcing Erdos-Hajnal graphs polarized partition relation Almost countably chromatic Mandelbrot set Lipschitz reduction Strongly compact cardinal Ineffable cardinal Partition Relations stick positive partition relation ccc Microscopic Approach
Tag Archives: 03E05
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
Putting a diamond inside the square
Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
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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
A cofinality-preserving small forcing may introduce a special Aronszajn tree
Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
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A relative of the approachability ideal, diamond and non-saturation
Abstract: Let $\lambda$ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often. In this paper, for a subset $S\subset\lambda^+$, a normal subideal of … Continue reading
On guessing generalized clubs at the successors of regulars
Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation … Continue reading
On the consistency strength of the Milner-Sauer conjecture
Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading
The Ostaszewski square, and homogeneous Souslin trees
Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property: For every sequence $\langle A_i\mid i<\lambda\rangle$ … Continue reading
Posted in Publications, Souslin Hypothesis, Squares and Diamonds
Tagged 03E05, 03E35, Club Guessing, Fat stationary set, Ostaszewski square, Souslin Tree
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