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