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