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