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