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