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