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