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Distributive tree Reflecting stationary set C-sequence Non-saturation Diamond for trees indecomposable filter Mandelbrot set 54G20 Kurepa Hypothesis b-scale Rainbow sets Filter reflection Successor of Regular Cardinal Knaster and friends Partition Relations weak square ccc Was Ulam right? diamond star Commutative cancellative semigroups Diamond-sharp Entangled linear order Successor of Singular Cardinal ZFC construction Universal Sequences Almost-disjoint family Sakurai's Bell inequality O-space Square-Brackets Partition Relations Chromatic number Foundations Forcing Singular cardinals combinatorics SNR specializable Souslin tree Diamond HOD Interval topology on trees Hindman's Theorem AIM forcing S-Space Cardinal Invariants Hedetniemi's conjecture P-Ideal Dichotomy Almost countably chromatic Rado's conjecture Erdos-Hajnal graphs Strong coloring reflection principles Fat stationary set unbounded function Dowker space Postprocessing function higher Baire space Greatly Mahlo sap Parameterized proxy principle transformations Strongly Luzin set Coherent tree Local Club Condensation. Microscopic Approach Singular cofinality full tree Prikry-type forcing Almost Souslin coloring number Closed coloring Whitehead Problem Antichain Strongly compact cardinal Subadditive Ostaszewski square Subtle cardinal Generalized Clubs Ineffable cardinal GMA super-Souslin tree Lipschitz reduction Club Guessing Cardinal function Shelah's Strong Hypothesis L-space Countryman line Ascending path Forcing Axioms Chang's conjecture Iterated forcing Aronszajn tree stationary reflection nonmeager set square stick Ramsey theory over partitions Axiom R weak diamond middle diamond Forcing with side conditions OCA Martin's Axiom Prevalent singular cardinals Subnormal ideal Ascent Path Amenable C-sequence Uniformly coherent Uniformization Generalized descriptive set theory Sigma-Prikry Singular Density perfectly normal Sierpinski's onto mapping principle Erdos Cardinal Cohen real Selective Ultrafilter Constructible Universe strongly bounded groups Minimal Walks tensor product graph Monotonically far Weakly compact cardinal Slim tree Nonspecial tree Hereditarily Lindelöf space countably metacompact polarized partition relation very good scale xbox Reduced Power Dushnik-Miller Ulam matrix Subtle tree property weak Kurepa tree Well-behaved magma Vanishing levels regressive Souslin tree Rock n' Roll Commutative projection system stationary hitting positive partition relation Respecting tree Intersection model Jonsson cardinal Poset approachability ideal Souslin Tree Luzin set Uniformly homogeneous Open Access Large Cardinals free Boolean algebra projective Boolean algebra club_AD Absoluteness Small forcing Partition relations for trees Analytic sets Precaliber incompactness Fast club PFA(S)[S] PFA Fodor-type reflection square principles free Souslin tree Knaster
Tag Archives: incompactness
11th Young Set Theory Workshop, June 2018
I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that … Continue reading
Posted in Invited Talks
Tagged Aronszajn tree, C-sequence, incompactness, Knaster, Minimal Walks, Postprocessing function, square
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MFO workshop in Set Theory, February 2017
I gave an invited talk at the Set Theory workshop in Obwerwolfach, February 2017. Talk Title: Coloring vs. Chromatic. Abstract: In a joint work with Chris Lambie-Hanson, we study the interaction between compactness for the chromatic number (of graphs) and … Continue reading
Posted in Invited Talks
Tagged Chromatic number, coloring number, incompactness, stationary reflection
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Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
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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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