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