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