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