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