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