Lower bounds on coloring numbers from hardness hypotheses in PCF theory

We prove that the statement “for every infinite cardinal κ, every graph with list-chromatic number κ has coloring number at most ℶ_ω (κ)” proved by Kojman (2014) using the RGCH theorem implies the WRGCH theorem, which is a weaker relative of the RGCH, via a short forcing argument. Similarly, a better upper bound than ℶ_ω (κ) in this statement implies stronger (consistent) forms of the WRGCH theorem, the consistency of whose negations is wide open. Thus, the optimality of Kojman’s upper bound is a purely cardinal arithmetic problem, and, as discussed below, is hard to decide.