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