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club_AD free Boolean algebra Reflecting stationary set AIM forcing Fodor-type reflection nonmeager set Sierpinski's onto mapping principle Fast club Countryman line P-Ideal Dichotomy Commutative cancellative semigroups Was Ulam right? Uniformly homogeneous Rock n' Roll weak diamond weak square PFA stationary reflection Universal Sequences Successor of Regular Cardinal incompactness Lipschitz reduction Commutative projection system Dowker space diamond star Strongly compact cardinal Subtle tree property Axiom R stationary hitting Uniformization Interval topology on trees Diamond-sharp Ascending path Rado's conjecture Prikry-type forcing approachability ideal specializable Souslin tree Forcing Axioms Kurepa Hypothesis Erdos Cardinal Vanishing levels Cohen real Sigma-Prikry xbox Whitehead Problem Successor of Singular Cardinal Ulam matrix tensor product graph square Mandelbrot set Singular cofinality Hereditarily Lindelöf space OCA Cardinal function Strongly Luzin set Ramsey theory over partitions Rainbow sets Chromatic number Monotonically far polarized partition relation countably metacompact Non-saturation Generalized Clubs Foundations 54G20 Hindman's Theorem full tree Analytic sets coloring number Weakly compact cardinal SNR Reduced Power Filter reflection higher Baire space Uniformly coherent very good scale Diamond Sakurai's Bell inequality unbounded function Poset Chang's conjecture Ostaszewski square strongly bounded groups Antichain Forcing with side conditions Intersection model Generalized descriptive set theory Entangled linear order Shelah's Strong Hypothesis Microscopic Approach Luzin set Absoluteness Respecting tree Hedetniemi's conjecture Ineffable cardinal Club Guessing Postprocessing function Almost-disjoint family Aronszajn tree Souslin Tree Minimal Walks perfectly normal Parameterized proxy principle regressive Souslin tree Ascent Path Local Club Condensation. stick projective Boolean algebra C-sequence Partition Relations GMA Forcing transformations Precaliber Singular cardinals combinatorics Prevalent singular cardinals indecomposable filter Jonsson cardinal Large Cardinals Knaster Singular Density Constructible Universe Erdos-Hajnal graphs Knaster and friends ccc Subadditive positive partition relation reflection principles Square-Brackets Partition Relations Martin's Axiom Cardinal Invariants weak Kurepa tree Subnormal ideal Diamond for trees Nonspecial tree Closed coloring Distributive tree square principles Greatly Mahlo Small forcing Dushnik-Miller Almost Souslin Almost countably chromatic b-scale Coherent tree HOD Open Access PFA(S)[S] free Souslin tree Iterated forcing sap O-space Strong coloring Slim tree ZFC construction L-space Fat stationary set super-Souslin tree S-Space Selective Ultrafilter Well-behaved magma Amenable C-sequence Subtle cardinal middle diamond
Tag Archives: 05C15
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
