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