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