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