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Sierpinski's onto mapping principle OCA xbox Parameterized proxy principle Subtle tree property Microscopic Approach Was Ulam right stationary reflection Ostaszewski square Square-Brackets Partition Relations Hedetniemi's conjecture Commutative cancellative semigroups full tree Ineffable cardinal club_AD Small forcing free Boolean algebra C-sequence indecomposable ultrafilter Fast club Forcing Axioms Well-behaved magma Reflecting stationary set 54G20 Absoluteness polarized partition relation Generalized descriptive set theory Luzin set Analytic sets stick PFA(S)[S] Uniformization Minimal Walks Vanishing levels nonmeager set O-space Nonspecial tree coloring number Singular cardinals combinatorics Poset weak square Lipschitz reduction GMA Strong coloring ccc Cohen real Fat stationary set Martin's Axiom Subtle cardinal Chang's conjecture Hereditarily Lindelöf space free Souslin tree Large Cardinals Constructible Universe Precaliber diamond star Dushnik-Miller Chromatic number Souslin Tree AIM forcing Successor of Singular Cardinal Generalized Clubs Greatly Mahlo approachability ideal Rainbow sets Antichain Distributive tree Rado's conjecture Almost Souslin Hindman's Theorem Universal Sequences Jonsson cardinal Diamond for trees Ulam matrix Almost-disjoint family Forcing Diamond-sharp Knaster Ascent Path projective Boolean algebra Sigma-Prikry Coherent tree Amenable C-sequence tensor product graph transformations Knaster and friends sap Singular Density b-scale Prikry-type forcing Selective Ultrafilter Successor of Regular Cardinal Foundations Subnormal ideal regressive Souslin tree square P-Ideal Dichotomy strongly bounded groups reflection principles Fodor-type reflection specializable Souslin tree higher Baire space Sakurai's Bell inequality Dowker space very good scale middle diamond S-Space Postprocessing function L-space Slim tree Almost countably chromatic weak Kurepa tree Uniformly coherent stationary hitting Aronszajn tree Kurepa Hypothesis Singular cofinality weak diamond PFA Club Guessing incompactness Mandelbrot set Weakly compact cardinal Local Club Condensation. Non-saturation Erdos-Hajnal graphs countably metacompact Subadditive Diamond Iterated forcing Filter reflection unbounded function Strongly Luzin set super-Souslin tree Cardinal Invariants Ramsey theory over partitions Prevalent singular cardinals positive partition relation Reduced Power Uniformly homogeneous Erdos Cardinal ZFC construction Rock n' Roll Open Access Axiom R Partition Relations Shelah's Strong Hypothesis square principles HOD SNR Cardinal function Closed coloring Whitehead Problem
Tag Archives: 03E05
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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