Archives
Keywords
strongly bounded groups Nonspecial tree PFA(S)[S] Filter reflection Greatly Mahlo Hindman's Theorem free Boolean algebra 54G20 Dowker space Large Cardinals coloring number countably metacompact indecomposable ultrafilter Almost Souslin Rainbow sets AIM forcing Commutative cancellative semigroups Luzin set P-Ideal Dichotomy S-Space Knaster and friends Prikry-type forcing Singular Density Small forcing regressive Souslin tree weak Kurepa tree Hedetniemi's conjecture HOD Constructible Universe Chromatic number Strongly Luzin set Generalized descriptive set theory Universal Sequences polarized partition relation unbounded function Successor of Regular Cardinal Slim tree Subtle tree property b-scale Absoluteness Closed coloring incompactness Singular cardinals combinatorics Cardinal function stationary reflection Reflecting stationary set club_AD Sakurai's Bell inequality Well-behaved magma Intersection model Strong coloring Erdos-Hajnal graphs Strongly compact cardinal Lipschitz reduction full tree Diamond for trees Chang's conjecture Successor of Singular Cardinal Ulam matrix Cohen real weak diamond Subadditive Cardinal Invariants Postprocessing function middle diamond sap Mandelbrot set Axiom R Minimal Walks square principles Selective Ultrafilter Distributive tree Dushnik-Miller Shelah's Strong Hypothesis Uniformization Ostaszewski square stick Square-Brackets Partition Relations Jonsson cardinal stationary hitting Ineffable cardinal Open Access Subnormal ideal approachability ideal L-space Fat stationary set Vanishing levels Aronszajn tree PFA specializable Souslin tree GMA C-sequence Microscopic Approach Rado's conjecture xbox Prevalent singular cardinals Reduced Power higher Baire space Amenable C-sequence Ascent Path Generalized Clubs Iterated forcing Whitehead Problem Analytic sets Martin's Axiom Fodor-type reflection Almost countably chromatic Fast club super-Souslin tree Poset very good scale reflection principles Club Guessing Erdos Cardinal Forcing Axioms Hereditarily Lindelöf space free Souslin tree Diamond tensor product graph ZFC construction OCA O-space Sigma-Prikry Forcing Partition Relations Weakly compact cardinal Antichain transformations Non-saturation Almost-disjoint family SNR Local Club Condensation. square Coherent tree Countryman line Was Ulam right diamond star Uniformly coherent positive partition relation Knaster Sierpinski's onto mapping principle Precaliber ccc nonmeager set Singular cofinality Ramsey theory over partitions Respecting tree projective Boolean algebra Commutative projection system weak square Subtle cardinal Souslin Tree Kurepa Hypothesis Foundations Diamond-sharp Parameterized proxy principle Rock n' Roll Uniformly homogeneous
Tag Archives: approachability ideal
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
Same Graph, Different Universe
Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading
Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
10 Comments
Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
Leave a comment
Shelah’s approachability ideal (part 1)
Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so … Continue reading
Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading
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
