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