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sap Hindman's Theorem stationary reflection Was Ulam right? Sierpinski's onto mapping principle countably metacompact Mandelbrot set Ostaszewski square Reduced Power Strongly compact cardinal Countryman line reflection principles Greatly Mahlo Hedetniemi's conjecture tensor product graph Iterated forcing ZFC construction Erdos Cardinal Hereditarily Lindelöf space Foundations higher Baire space Generalized Clubs Almost countably chromatic Reflecting stationary set Successor of Regular Cardinal ccc transformations Club Guessing Shelah's Strong Hypothesis Cohen real Kurepa Hypothesis Ulam matrix Intersection model Knaster Knaster and friends Martin's Axiom Subnormal ideal nonmeager set Filter reflection OCA Large Cardinals strongly bounded groups stationary hitting Erdos-Hajnal graphs S-Space Diamond for trees Microscopic Approach Subadditive Small forcing positive partition relation Partition Relations Rock n' Roll regressive Souslin tree HOD Diamond square indecomposable filter Singular cardinals combinatorics square principles Prikry-type forcing Subtle tree property Chromatic number Postprocessing function Jonsson cardinal 54G20 Generalized descriptive set theory Coherent tree Uniformly homogeneous Ascending path full tree Dushnik-Miller xbox Fat stationary set Absoluteness Partition relations for trees Singular Density Luzin set incompactness Entangled linear order weak Kurepa tree Nonspecial tree Forcing with side conditions Antichain Poset PFA Aronszajn tree free Boolean algebra super-Souslin tree Forcing Axioms projective Boolean algebra Sigma-Prikry Square-Brackets Partition Relations Diamond-sharp Successor of Singular Cardinal Analytic sets SNR Vanishing levels club_AD very good scale Almost Souslin Parameterized proxy principle Cardinal Invariants Chang's conjecture stick Lipschitz reduction GMA Axiom R Fodor-type reflection Local Club Condensation. Cardinal function Sakurai's Bell inequality Amenable C-sequence Minimal Walks PFA(S)[S] Constructible Universe Precaliber coloring number polarized partition relation O-space Rainbow sets Slim tree Fast club Whitehead Problem Commutative cancellative semigroups Souslin Tree Prevalent singular cardinals Non-saturation Ineffable cardinal Rado's conjecture Dowker space Strongly Luzin set perfectly normal Respecting tree Open Access middle diamond b-scale Subtle cardinal specializable Souslin tree L-space Selective Ultrafilter Strong coloring Forcing Monotonically far C-sequence Uniformization approachability ideal Almost-disjoint family Distributive tree P-Ideal Dichotomy Ascent Path diamond star free Souslin tree Weakly compact cardinal Commutative projection system Well-behaved magma Universal Sequences weak square AIM forcing Uniformly coherent Singular cofinality weak diamond Closed coloring Interval topology on trees unbounded function Ramsey theory over partitions
Author Archives: Assaf Rinot
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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On topological spaces of singular density and minimal weight
Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading
Shelah’s solution to Whitehead’s problem
Whitehead problem notes in hebrew : Table of contents Chapter 0 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 References
