### 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)
in
this statement implies stronger forms of the RGCH theorem whose
consistency as well as the consistency of their negations are
wide
open.
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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