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