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Reflecting stationary set Strong coloring free Boolean algebra higher Baire space Cardinal Invariants Prevalent singular cardinals Foundations C-sequence OCA Diamond-sharp Filter reflection Ineffable cardinal PFA(S)[S] Fat stationary set stick Uniformly coherent Dushnik-Miller Non-saturation Chromatic number specializable Souslin tree Sierpinski's onto mapping principle Subtle tree property Partition Relations Ascent Path Slim tree super-Souslin tree Lipschitz reduction Almost countably chromatic Was Ulam right Commutative cancellative semigroups Club Guessing Distributive tree Knaster Intersection model Minimal Walks GMA Large Cardinals Singular Density Kurepa Hypothesis full tree sap Vanishing levels square Rainbow sets Generalized Clubs Souslin Tree Ramsey theory over partitions L-space P-Ideal Dichotomy Generalized descriptive set theory xbox ZFC construction Sakurai's Bell inequality Successor of Regular Cardinal Whitehead Problem Uniformly homogeneous strongly bounded groups Shelah's Strong Hypothesis Reduced Power Poset Closed coloring Nonspecial tree Square-Brackets Partition Relations transformations stationary reflection Cohen real Postprocessing function Sigma-Prikry Erdos Cardinal Small forcing Dowker space tensor product graph reflection principles Fast club Countryman line Weakly compact cardinal b-scale club_AD Well-behaved magma Forcing Commutative projection system Local Club Condensation. Coherent tree projective Boolean algebra Fodor-type reflection Microscopic Approach Greatly Mahlo Singular cofinality indecomposable ultrafilter Almost-disjoint family Parameterized proxy principle Luzin set Open Access Subtle cardinal free Souslin tree PFA Mandelbrot set Erdos-Hajnal graphs Amenable C-sequence Prikry-type forcing ccc weak square 54G20 SNR stationary hitting Ulam matrix Rado's conjecture Subadditive weak Kurepa tree weak diamond Almost Souslin Uniformization Diamond for trees S-Space Martin's Axiom Respecting tree very good scale Constructible Universe regressive Souslin tree Forcing Axioms Aronszajn tree Hindman's Theorem nonmeager set Knaster and friends Antichain square principles HOD Singular cardinals combinatorics Absoluteness Hedetniemi's conjecture Strongly compact cardinal Iterated forcing incompactness Precaliber Chang's conjecture Successor of Singular Cardinal countably metacompact middle diamond diamond star Strongly Luzin set Cardinal function AIM forcing Jonsson cardinal Rock n' Roll coloring number Subnormal ideal Selective Ultrafilter Universal Sequences Axiom R Analytic sets approachability ideal Hereditarily Lindelöf space O-space polarized partition relation unbounded function positive partition relation Diamond Ostaszewski square
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 $Vspec(\kappa)$ of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa$-tree $\mathbf T$. The latter is an invariant in … Continue reading
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
