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