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