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