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Subadditive Reduced Power middle diamond 54G20 Selective Ultrafilter Subnormal ideal Weakly compact cardinal higher Baire space square free Boolean algebra Prevalent singular cardinals Diamond Absoluteness Filter reflection free Souslin tree Countryman line Ramsey theory over partitions Strongly Luzin set full tree transformations Chang's conjecture Fast club Singular cofinality Fat stationary set Interval topology on trees Was Ulam right? Forcing with side conditions Subtle tree property Ascending path L-space Minimal Walks Small forcing sap Erdos-Hajnal graphs Iterated forcing Amenable C-sequence Commutative cancellative semigroups Successor of Regular Cardinal Slim tree Almost Souslin coloring number Hindman's Theorem Intersection model Rainbow sets Respecting tree Generalized Clubs polarized partition relation Erdos Cardinal Foundations Sakurai's Bell inequality Local Club Condensation. Ineffable cardinal Cohen real stationary reflection square principles b-scale weak Kurepa tree Axiom R Mandelbrot set Souslin Tree C-sequence tensor product graph Square-Brackets Partition Relations unbounded function Dushnik-Miller Postprocessing function Large Cardinals weak square Reflecting stationary set Singular cardinals combinatorics Uniformly homogeneous Whitehead Problem Universal Sequences Ascent Path Antichain Successor of Singular Cardinal Sigma-Prikry Uniformly coherent Almost countably chromatic Almost-disjoint family Kurepa Hypothesis OCA reflection principles Partition Relations Commutative projection system Analytic sets stationary hitting Chromatic number positive partition relation club_AD Martin's Axiom Rock n' Roll Closed coloring Jonsson cardinal P-Ideal Dichotomy perfectly normal Precaliber Prikry-type forcing strongly bounded groups Knaster very good scale Ostaszewski square AIM forcing Sierpinski's onto mapping principle stick Coherent tree Hereditarily Lindelöf space Nonspecial tree S-Space Greatly Mahlo nonmeager set Poset Fodor-type reflection Generalized descriptive set theory SNR Dowker space Diamond for trees HOD incompactness regressive Souslin tree ZFC construction projective Boolean algebra Forcing Axioms Distributive tree O-space weak diamond Rado's conjecture Singular Density Monotonically far xbox PFA Non-saturation Entangled linear order Club Guessing indecomposable filter Ulam matrix super-Souslin tree GMA Shelah's Strong Hypothesis Subtle cardinal Lipschitz reduction PFA(S)[S] Strong coloring approachability ideal Uniformization Knaster and friends countably metacompact Hedetniemi's conjecture Luzin set Parameterized proxy principle Aronszajn tree Constructible Universe Open Access Strongly compact cardinal Forcing Diamond-sharp diamond star Well-behaved magma Vanishing levels Cardinal function specializable Souslin tree Partition relations for trees Microscopic Approach Cardinal Invariants ccc
Category Archives: Squares and Diamonds
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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The failure of diamond on a reflecting stationary set
Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading
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
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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