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