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