  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
[6] using the RGCH theorem [11] implies the RGCG theorem via a short forcing
argument. Similarly, a better upper bound than beth_omega (nu) in this
statement implies stronger forms of the RGCH theorem hold, whose consistency
and the consistency of their negations are wide open. Thus, the optimality of
Kojman's upper bound is a purely cardinal arithmetic problem, and, as discussed
below, is hard to decide.

Logic