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Absoluteness Slim tree Analytic sets stationary hitting stationary reflection Ulam matrix OCA tensor product graph L-space Sigma-Prikry Precaliber stick Respecting tree ZFC construction transformations Dowker space sap Strongly compact cardinal Fat stationary set Partition Relations HOD Large Cardinals Diamond diamond star Martin's Axiom square principles regressive Souslin tree Strongly Luzin set Successor of Regular Cardinal projective Boolean algebra coloring number Knaster and friends Shelah's Strong Hypothesis Subtle cardinal Amenable C-sequence Singular cofinality Open Access reflection principles Erdos Cardinal Monotonically far Luzin set Reduced Power full tree approachability ideal Ascent Path Postprocessing function Prevalent singular cardinals square Rainbow sets Parameterized proxy principle Cohen real Coherent tree Almost Souslin Generalized descriptive set theory SNR Local Club Condensation. ccc C-sequence Was Ulam right? club_AD Antichain Vanishing levels polarized partition relation Mandelbrot set Interval topology on trees weak diamond Strong coloring S-Space Poset Microscopic Approach Ascending path Ramsey theory over partitions Diamond for trees 54G20 Reflecting stationary set Cardinal Invariants Diamond-sharp Club Guessing Axiom R Singular Density Partition relations for trees Aronszajn tree Nonspecial tree very good scale positive partition relation Small forcing Uniformly homogeneous indecomposable filter Forcing Axioms Subnormal ideal Cardinal function Selective Ultrafilter Hereditarily Lindelöf space Chromatic number perfectly normal free Boolean algebra Sakurai's Bell inequality specializable Souslin tree weak square Hindman's Theorem unbounded function Kurepa Hypothesis xbox Commutative projection system Iterated forcing Jonsson cardinal Almost countably chromatic free Souslin tree middle diamond Whitehead Problem Uniformization Subadditive O-space Square-Brackets Partition Relations Subtle tree property Singular cardinals combinatorics Sierpinski's onto mapping principle AIM forcing PFA super-Souslin tree Uniformly coherent Countryman line Fodor-type reflection Fast club Almost-disjoint family Rock n' Roll Successor of Singular Cardinal Filter reflection nonmeager set weak Kurepa tree Greatly Mahlo Non-saturation Minimal Walks Forcing with side conditions Foundations Knaster Souslin Tree Entangled linear order Rado's conjecture GMA Chang's conjecture Prikry-type forcing Commutative cancellative semigroups Distributive tree Hedetniemi's conjecture Closed coloring Ineffable cardinal Erdos-Hajnal graphs b-scale Constructible Universe Lipschitz reduction incompactness Weakly compact cardinal Intersection model Ostaszewski square countably metacompact strongly bounded groups Generalized Clubs Well-behaved magma Forcing P-Ideal Dichotomy PFA(S)[S] higher Baire space Dushnik-Miller Universal Sequences
Tag Archives: Almost-disjoint family
The vanishing levels of a tree
Joint work with Shira Yadai and Zhixing You. Abstract. We initiate the study of the spectrum of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa$-tree $\mathbf T$. This is an invariant in the … Continue reading
Posted in Preprints, Souslin Hypothesis
Tagged Almost-disjoint family, Ascent Path, C-sequence, Coherent tree, Dowker space, Open Access, Parameterized proxy principle, regressive Souslin tree, Respecting tree, Subtle tree property, Uniformly homogeneous, Vanishing levels, weak Kurepa tree
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Universal binary sequences
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Suppose for the moment that we are given a fixed sequence $\langle f_\alpha:\omega\rightarrow2\mid \alpha\in a\rangle$, indexed by some set $a$ of ordinals. Then, for every function $h:a\rightarrow\omega$ and $i<\omega$, we … Continue reading
The Engelking-Karlowicz theorem, and a useful corollary
Theorem (Engelking-Karlowicz, 1965). For cardinals $\kappa\le\lambda\le\mu\le 2^\lambda$, the following are equivalent: $\lambda^{<\kappa}=\lambda$; there exists a collection of functions, $\langle f_i:\mu\rightarrow\lambda\mid i<\lambda\rangle$, such that for every $X\in[\mu]^{<\kappa}$ and every function $f:X\rightarrow\lambda$, there exists some $i<\lambda$ with $f\subseteq f_i$. Proof. (2)$\Rightarrow$(1) Suppose … Continue reading
