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