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