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