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