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