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