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