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