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