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Minimal Walks weak Kurepa tree Well-behaved magma Forcing with side conditions Weakly compact cardinal Fast club Subtle tree property Jonsson cardinal Club Guessing Rainbow sets Lipschitz reduction Dushnik-Miller Cohen real Sierpinski's onto mapping principle incompactness free Souslin tree square Chang's conjecture Reduced Power S-Space nonmeager set Knaster Almost countably chromatic Intersection model middle diamond stationary hitting Constructible Universe PFA Iterated forcing Poset Shelah's Strong Hypothesis stationary reflection Hindman's Theorem weak diamond Almost-disjoint family full tree Martin's Axiom Entangled linear order sap Ramsey theory over partitions Subadditive OCA Uniformization Sigma-Prikry Partition relations for trees Vanishing levels regressive Souslin tree Was Ulam right? free Boolean algebra Slim tree Coherent tree higher Baire space HOD projective Boolean algebra Uniformly homogeneous approachability ideal Singular Density Generalized descriptive set theory Reflecting stationary set Erdos-Hajnal graphs Prikry-type forcing Interval topology on trees Luzin set Chromatic number coloring number Generalized Clubs Mandelbrot set club_AD Small forcing Ostaszewski square P-Ideal Dichotomy Singular cardinals combinatorics Commutative projection system Almost Souslin Forcing Axioms Strongly Luzin set reflection principles Foundations perfectly normal O-space tensor product graph unbounded function Non-saturation Fodor-type reflection Knaster and friends Closed coloring Successor of Regular Cardinal Hereditarily Lindelöf space weak square Strong coloring Successor of Singular Cardinal Greatly Mahlo strongly bounded groups Diamond for trees SNR Strongly compact cardinal Precaliber Aronszajn tree Rado's conjecture Antichain Uniformly coherent Amenable C-sequence Diamond-sharp Kurepa Hypothesis Fat stationary set Ascending path Diamond super-Souslin tree very good scale Universal Sequences stick Whitehead Problem Filter reflection Large Cardinals Postprocessing function Nonspecial tree xbox AIM forcing C-sequence Ineffable cardinal ZFC construction Forcing Sakurai's Bell inequality positive partition relation polarized partition relation countably metacompact Absoluteness Distributive tree square principles diamond star Selective Ultrafilter Partition Relations Hedetniemi's conjecture Ulam matrix Respecting tree Cardinal function Parameterized proxy principle transformations Prevalent singular cardinals ccc Axiom R Open Access Erdos Cardinal Square-Brackets Partition Relations Monotonically far 54G20 GMA L-space Rock n' Roll b-scale Local Club Condensation. Microscopic Approach Analytic sets Singular cofinality Souslin Tree Subnormal ideal PFA(S)[S] Ascent Path Commutative cancellative semigroups Dowker space Subtle cardinal specializable Souslin tree Cardinal Invariants indecomposable filter Countryman line
Category Archives: Compactness
Chromatic numbers of graphs – large gaps
Abstract. We say that a graph $G$ is $(\aleph_0,\kappa)$-chromatic if $\text{Chr}(G)=\kappa$, while $\text{Chr}(G’)\le\aleph_0$ for any subgraph $G’$ of $G$ of size $<|G|$. The main result of this paper reads as follows. If $\square_\lambda+\text{CH}_\lambda$ holds for a given uncountable cardinal $\lambda$, … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Almost countably chromatic, Chromatic number, incompactness, Ostaszewski square
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A topological reflection principle equivalent to Shelah’s strong hypothesis
Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading
Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Open Access, Shelah's Strong Hypothesis
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Openly generated Boolean algebras and the Fodor-type reflection principle
Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading
Posted in Compactness, Publications
Tagged 03E35, 03E55, 03E65, 03E75, 03G05, 06E05, Axiom R, Fodor-type reflection, free Boolean algebra, projective Boolean algebra, Shelah's Strong Hypothesis, stationary reflection
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