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Filter reflection Sakurai's Bell inequality super-Souslin tree Shelah's Strong Hypothesis Ascending path weak square Local Club Condensation. Reduced Power Dowker space Precaliber SNR positive partition relation Open Access Generalized Clubs Entangled linear order Successor of Singular Cardinal Erdos-Hajnal graphs coloring number S-Space Slim tree perfectly normal Universal Sequences Jonsson cardinal square P-Ideal Dichotomy Almost-disjoint family Non-saturation stationary hitting projective Boolean algebra Respecting tree Ascent Path countably metacompact Ulam matrix Subnormal ideal diamond star Sierpinski's onto mapping principle Aronszajn tree Kurepa Hypothesis Selective Ultrafilter Knaster and friends Cohen real Uniformization O-space Ramsey theory over partitions very good scale indecomposable filter C-sequence Reflecting stationary set Strong coloring HOD Diamond for trees GMA Monotonically far Commutative projection system club_AD Uniformly homogeneous specializable Souslin tree Chang's conjecture Large Cardinals Diamond Greatly Mahlo Almost Souslin Lipschitz reduction Knaster approachability ideal OCA Rado's conjecture Erdos Cardinal Distributive tree Prevalent singular cardinals weak diamond Minimal Walks Interval topology on trees Antichain Analytic sets Small forcing Partition Relations Almost countably chromatic incompactness Vanishing levels xbox middle diamond Souslin Tree Prikry-type forcing Singular cofinality Commutative cancellative semigroups Was Ulam right? stationary reflection Subtle tree property Amenable C-sequence ZFC construction Forcing Poset Hindman's Theorem ccc Strongly Luzin set Weakly compact cardinal regressive Souslin tree Nonspecial tree Luzin set Countryman line Singular Density higher Baire space sap b-scale nonmeager set Foundations Iterated forcing AIM forcing PFA(S)[S] Square-Brackets Partition Relations Postprocessing function Whitehead Problem Fodor-type reflection Uniformly coherent 54G20 Cardinal Invariants L-space full tree Cardinal function Closed coloring Singular cardinals combinatorics Absoluteness Axiom R Forcing with side conditions weak Kurepa tree free Boolean algebra Microscopic Approach Parameterized proxy principle tensor product graph Subtle cardinal transformations free Souslin tree Ineffable cardinal Fat stationary set Intersection model strongly bounded groups Sigma-Prikry Rock n' Roll Hedetniemi's conjecture Rainbow sets Subadditive Forcing Axioms Coherent tree stick Well-behaved magma Partition relations for trees Ostaszewski square Strongly compact cardinal reflection principles Dushnik-Miller Generalized descriptive set theory Mandelbrot set Successor of Regular Cardinal PFA unbounded function Martin's Axiom Club Guessing Fast club Hereditarily Lindelöf space Diamond-sharp polarized partition relation Constructible Universe square principles Chromatic number
Author Archives: Assaf Rinot
Dushnik-Miller for regular cardinals (part 2)
In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading
Posted in Blog, Expository
Tagged b-scale, Dushnik-Miller, Partition Relations, Square-Brackets Partition Relations
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Dushnik-Miller for singular cardinals (part 1)
Continuing the previous post, let us now prove the following. Theorem (Erdos-Dushnik-Miller, 1941). For every singular cardinal λ, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Proof. Suppose that $\lambda$ is a singular cardinal, and $c:[\lambda]^2\rightarrow\{0,1\}$ is a given coloring. For any ordinal $\alpha<\lambda$, denote … Continue reading
Dushnik-Miller for regular cardinals (part 1)
This is the first out of a series of posts on the following theorem. Theorem (Erdos-Dushnik-Miller, 1941). For every infinite cardinal $\lambda$, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Namely, for any coloring $c:[\lambda]^2\rightarrow\{0,1\}$ there exists either a subset $A\subseteq \lambda$ of order-type $\lambda$ with … Continue reading
The order-type of clubs in a square sequence
Recall Jensen’s notion of square: Definition (Jensen): For an infinite cardinal $\lambda$, $\square_\lambda$ asserts the existence of a sequence $\overrightarrow C=\left\langle C_\alpha\mid\alpha\in\text{acc}(\lambda^+)\right\rangle$ such that for every limit $\alpha<\lambda^+$: $C_\alpha$ is a club subset of $\alpha$ of order-type $\le\lambda$; if $\beta\in\text{acc}(C_\alpha)$, … Continue reading
Music Video: “Wide Open” by Jenny Mayhem
Did you notice the toolbar at the bottom of my posts? e.g.:
Posted in Blog, OffMath
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The search for diamonds
Abstract: This is a review I wrote for the Bulletin of Symbolic Logic on the following papers: Saharon Shelah, Middle Diamond, Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah, Diamonds, Proceedings of the American Mathematical Society, vol. … Continue reading
Posted in Publications, Reviews, Squares and Diamonds
Tagged Diamond, middle diamond, weak diamond, weak square
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Aspects of singular cofinality
Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer … 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
A cofinality-preserving small forcing may introduce a special Aronszajn tree
Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
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