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weak square Subtle cardinal Cohen real Rock n' Roll ZFC construction Diamond for trees Almost Souslin Distributive tree incompactness full tree Was Ulam right? Knaster Hindman's Theorem Postprocessing function Ascending path Antichain club_AD polarized partition relation xbox PFA(S)[S] Slim tree Club Guessing P-Ideal Dichotomy Forcing Hedetniemi's conjecture Countryman line C-sequence 54G20 regressive Souslin tree Forcing Axioms Minimal Walks square principles Small forcing stationary reflection Uniformization Analytic sets perfectly normal Rainbow sets Vanishing levels Reduced Power Interval topology on trees Selective Ultrafilter Iterated forcing Jonsson cardinal Microscopic Approach Precaliber Luzin set Non-saturation reflection principles Almost-disjoint family Diamond Axiom R Closed coloring GMA super-Souslin tree Subnormal ideal transformations stick Fat stationary set Sigma-Prikry Greatly Mahlo AIM forcing b-scale Poset sap L-space weak Kurepa tree Mandelbrot set Diamond-sharp Large Cardinals square Ramsey theory over partitions Uniformly coherent Generalized descriptive set theory Singular Density Martin's Axiom Fodor-type reflection very good scale tensor product graph unbounded function Erdos Cardinal Parameterized proxy principle Respecting tree strongly bounded groups SNR Kurepa Hypothesis PFA ccc Whitehead Problem weak diamond free Boolean algebra Constructible Universe nonmeager set free Souslin tree Open Access Prevalent singular cardinals Amenable C-sequence Partition Relations Chang's conjecture Monotonically far Square-Brackets Partition Relations Local Club Condensation. Souslin Tree approachability ideal Commutative cancellative semigroups Fast club Strong coloring Filter reflection Nonspecial tree Weakly compact cardinal Generalized Clubs indecomposable filter Successor of Regular Cardinal Ostaszewski square Knaster and friends Partition relations for trees Prikry-type forcing Sakurai's Bell inequality countably metacompact Absoluteness Subtle tree property Foundations Well-behaved magma Erdos-Hajnal graphs Singular cardinals combinatorics Ulam matrix Uniformly homogeneous Intersection model Lipschitz reduction Ascent Path Strongly Luzin set S-Space Dushnik-Miller higher Baire space specializable Souslin tree Entangled linear order Sierpinski's onto mapping principle Commutative projection system positive partition relation Chromatic number middle diamond Coherent tree Forcing with side conditions Reflecting stationary set Hereditarily Lindelöf space Subadditive Cardinal function OCA O-space Aronszajn tree stationary hitting Rado's conjecture coloring number Singular cofinality Ineffable cardinal diamond star Cardinal Invariants HOD Shelah's Strong Hypothesis Almost countably chromatic Dowker space Universal Sequences Strongly compact cardinal projective Boolean algebra Successor of Singular Cardinal
Tag Archives: 03E35
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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Openly generated Boolean algebras and the Fodor-type reflection principle
Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading
Posted in Compactness, Publications
Tagged 03E35, 03E55, 03E65, 03E75, 03G05, 06E05, Axiom R, Fodor-type reflection, free Boolean algebra, projective Boolean algebra, Shelah's Strong Hypothesis, stationary reflection
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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
Antichains in partially ordered sets of singular cofinality
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 main result of of this … Continue reading
Posted in Publications, Singular Cardinals Combinatorics
Tagged 03E04, 03E35, 06A07, Antichain, Poset, Singular cofinality
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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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