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countably metacompact Dowker space Antichain coloring number Minimal Walks Strong coloring Entangled linear order Strongly Luzin set Ineffable cardinal C-sequence PFA very good scale Selective Ultrafilter Weakly compact cardinal Chromatic number Subtle cardinal nonmeager set Club Guessing full tree Analytic sets Sigma-Prikry Lipschitz reduction weak Kurepa tree Local Club Condensation. free Souslin tree Aronszajn tree AIM forcing Greatly Mahlo specializable Souslin tree Hedetniemi's conjecture Shelah's Strong Hypothesis super-Souslin tree Forcing Axioms Sierpinski's onto mapping principle stationary hitting incompactness Rock n' Roll Reduced Power ccc Hindman's Theorem Erdos-Hajnal graphs Small forcing strongly bounded groups PFA(S)[S] Subadditive Diamond for trees Fodor-type reflection projective Boolean algebra Precaliber Chang's conjecture Foundations Almost-disjoint family Ramsey theory over partitions Reflecting stationary set Was Ulam right? Nonspecial tree weak diamond positive partition relation Cohen real free Boolean algebra Fast club Almost countably chromatic Axiom R Prikry-type forcing Subtle tree property Monotonically far Generalized Clubs Kurepa Hypothesis Diamond-sharp Uniformly coherent O-space middle diamond square Ostaszewski square Generalized descriptive set theory Slim tree Cardinal function Partition relations for trees club_AD HOD Interval topology on trees perfectly normal Forcing GMA Sakurai's Bell inequality diamond star Fat stationary set Hereditarily Lindelöf space square principles Non-saturation Large Cardinals Knaster and friends Rainbow sets Partition Relations Well-behaved magma P-Ideal Dichotomy regressive Souslin tree polarized partition relation Respecting tree Parameterized proxy principle Martin's Axiom stationary reflection Filter reflection Ulam matrix Commutative projection system Successor of Regular Cardinal Commutative cancellative semigroups ZFC construction Cardinal Invariants Erdos Cardinal Diamond Dushnik-Miller Luzin set transformations Square-Brackets Partition Relations L-space Knaster Postprocessing function Countryman line Forcing with side conditions indecomposable filter Uniformly homogeneous S-Space Iterated forcing unbounded function Coherent tree Vanishing levels xbox Universal Sequences Absoluteness Subnormal ideal Constructible Universe OCA sap Rado's conjecture b-scale Souslin Tree Distributive tree Almost Souslin higher Baire space Prevalent singular cardinals SNR Intersection model Strongly compact cardinal 54G20 Poset Jonsson cardinal approachability ideal Amenable C-sequence Singular cofinality Open Access tensor product graph Successor of Singular Cardinal Whitehead Problem stick Singular Density reflection principles Ascent Path Ascending path Closed coloring weak square Uniformization Singular cardinals combinatorics Microscopic Approach Mandelbrot set
Tag Archives: 05C63
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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Same Graph, Different Universe
Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading
Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
10 Comments
Hedetniemi’s conjecture for uncountable graphs
Abstract. It is proved that in Godel’s constructible universe, for every successor cardinal $\kappa$, there exist graphs $\mathcal G$ and $\mathcal H$ of size and chromatic number $\kappa$, for which the tensor product graph $\mathcal G\times\mathcal H$ is countably chromatic. … Continue reading
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
6 Comments
