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