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