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