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