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