Lower bounds on coloring numbers from hardness hypotheses in PCF theory
by Shelah. [Sh:1052]
Proc American Math Soc, 2016
We prove that the statement ``for every infinite cardinal nu,
every graph with list chromatic nu has coloring number at most
beth_omega (nu)'' proved by Kojman [koj] using the RGCH theorem
[sh:460] implies the RGCG theorem via a short forcing argument.
By the same method, a better upper bound than beth_omega (nu)
this statement implies stronger forms of the RGCH theorem whose
consistency as well as the consistency of their negations are
Thus, the optimality of Kojman's upper bound is
a purely cardinal arithmetic problem, which, as discussed below,
may be quite hard to decide.
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