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