Archives
Keywords
Rainbow sets 54G20 free Souslin tree higher Baire space Foundations polarized partition relation Well-behaved magma Sierpinski's onto mapping principle Partition Relations Fast club Dushnik-Miller perfectly normal middle diamond Singular Density Respecting tree approachability ideal PFA(S)[S] transformations Subadditive weak square Analytic sets nonmeager set Ascending path Subtle tree property Parameterized proxy principle L-space Countryman line Aronszajn tree stick Weakly compact cardinal C-sequence Non-saturation sap xbox P-Ideal Dichotomy Iterated forcing GMA Prikry-type forcing Strong coloring SNR countably metacompact Uniformization Erdos Cardinal Martin's Axiom positive partition relation Small forcing Microscopic Approach Closed coloring Knaster Fat stationary set Cardinal Invariants Forcing Square-Brackets Partition Relations Rado's conjecture super-Souslin tree Club Guessing Ostaszewski square incompactness Poset regressive Souslin tree Lipschitz reduction full tree Ulam matrix Postprocessing function Selective Ultrafilter Singular cofinality Strongly Luzin set square Local Club Condensation. HOD Monotonically far AIM forcing Generalized descriptive set theory Nonspecial tree Hedetniemi's conjecture Successor of Regular Cardinal Greatly Mahlo Singular cardinals combinatorics weak Kurepa tree Strongly compact cardinal Knaster and friends S-Space square principles reflection principles Reduced Power ZFC construction indecomposable filter Vanishing levels Constructible Universe diamond star Ramsey theory over partitions Almost-disjoint family O-space Jonsson cardinal Prevalent singular cardinals Large Cardinals Generalized Clubs Rock n' Roll Chang's conjecture Diamond weak diamond Slim tree Axiom R Dowker space Filter reflection Minimal Walks Chromatic number Hindman's Theorem Hereditarily Lindelöf space Ascent Path Antichain Open Access Precaliber ccc tensor product graph Almost countably chromatic Diamond for trees Interval topology on trees Subnormal ideal Mandelbrot set Commutative projection system stationary hitting very good scale Was Ulam right? Forcing with side conditions Subtle cardinal Luzin set coloring number Intersection model Universal Sequences Almost Souslin Distributive tree Whitehead Problem Uniformly homogeneous Fodor-type reflection projective Boolean algebra PFA Commutative cancellative semigroups specializable Souslin tree Ineffable cardinal club_AD Coherent tree Entangled linear order Partition relations for trees strongly bounded groups stationary reflection Shelah's Strong Hypothesis Souslin Tree OCA Uniformly coherent Cohen real Amenable C-sequence Kurepa Hypothesis Absoluteness Sakurai's Bell inequality free Boolean algebra Cardinal function Sigma-Prikry Erdos-Hajnal graphs b-scale Forcing Axioms Diamond-sharp Reflecting stationary set unbounded function Successor of Singular Cardinal
Tag Archives: Iterated forcing
Diamond on Kurepa trees
Joint work with Ziemek Kostana and Saharon Shelah. Abstract. We introduce a new weak variation of diamond that is meant to guess only the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by … Continue reading
Sigma-Prikry forcing III: Down to Aleph_omega
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classical results of … Continue reading
Ramsey theory over partitions III: Strongly Luzin sets and partition relations
Joint work with Menachem Kojman and Juris Steprāns. Abstract. The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of … Continue reading
Sigma-Prikry forcing II: Iteration Scheme
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\Sigma$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers … Continue reading
The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
2 Comments
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
