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