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