Abstract. We say that a graph
The main result of this paper reads as follows.
If
We also study
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Citation information:
A. Rinot, Chromatic numbers of graphs – large gaps, Combinatorica, 35(2): 215-233, 2015.
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Keywords
regressive Souslin tree free Souslin tree Ramsey theory over partitions Subtle tree property stationary reflection Small forcing Distributive tree Luzin set Successor of Regular Cardinal Dushnik-Miller Uniformization Coherent tree SNR nonmeager set very good scale club_AD Postprocessing function diamond star Subadditive stationary hitting Microscopic Approach Hindman's Theorem reflection principles Club Guessing weak square GMA Sierpinski's onto mapping principle Uniformly coherent Jonsson cardinal square Analytic sets Almost Souslin indecomposable ultrafilter Respecting tree ZFC construction Slim tree Diamond-sharp Reflecting stationary set stick Well-behaved magma Lipschitz reduction C-sequence Subtle cardinal Open Access full tree Generalized descriptive set theory L-space Commutative projection system countably metacompact Ostaszewski square strongly bounded groups Prevalent singular cardinals Diamond for trees unbounded function Cardinal Invariants Local Club Condensation. Ineffable cardinal Hereditarily Lindelöf space PFA Closed coloring Precaliber middle diamond Fat stationary set Antichain Intersection model Ascent Path Chromatic number Successor of Singular Cardinal Strong coloring Poset Foundations Shelah's Strong Hypothesis sap Non-saturation Rado's conjecture transformations Vanishing levels Cardinal function Mandelbrot set incompactness Singular cardinals combinatorics xbox Reduced Power Singular Density Singular cofinality Subnormal ideal OCA Rainbow sets Universal Sequences Selective Ultrafilter O-space Souslin Tree Amenable C-sequence Filter reflection Fast club Greatly Mahlo Minimal Walks Sakurai's Bell inequality approachability ideal Prikry-type forcing Erdos Cardinal Strongly compact cardinal Erdos-Hajnal graphs Whitehead Problem super-Souslin tree Knaster and friends higher Baire space AIM forcing Knaster Almost-disjoint family Forcing Axioms Axiom R Strongly Luzin set Partition Relations Ulam matrix Square-Brackets Partition Relations PFA(S)[S] Was Ulam right? Absoluteness coloring number projective Boolean algebra Constructible Universe Large Cardinals Chang's conjecture Aronszajn tree Nonspecial tree 54G20 weak diamond Cohen real ccc Iterated forcing Forcing free Boolean algebra positive partition relation Hedetniemi's conjecture weak Kurepa tree Fodor-type reflection S-Space tensor product graph Dowker space Martin's Axiom Parameterized proxy principle Generalized Clubs P-Ideal Dichotomy Diamond b-scale Sigma-Prikry Kurepa Hypothesis specializable Souslin tree Rock n' Roll Commutative cancellative semigroups Almost countably chromatic Weakly compact cardinal square principles HOD polarized partition relation Countryman line Uniformly homogeneous
That really is a simple method! Nice one Assaf!
Thank you very much!!
Submitted to Combinatorica, January 2013.
Accepted, April 2013.
Hi Assaf. It is a nice paper! (By the way, none of the references I looked at had anything as detailed as your results on changing chromatic numbers.)
Thanks a lot for checking!
p.s.
In addition to the results I told you about, I found more, including a ZFC example of an uncountably chromatic graph that can be made countably chromatic via a c.c.c. forcing, and that GCH entails analogous statements for successors of regulars above
It turns out there is no need for a nonreflecting stationary set!
In an upcoming paper with Chris Lambie-Hanson, we shall show that moreover the existence of an almost countably chromatic graph of size and chromatic number