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