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