Lower bounds on coloring numbers from hardness hypotheses in PCF theory
Abstract
We prove that the statement "for every infinite cardinal nu, every graph with list chromatic nu has coloring number at most bethomega (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 bethomega (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.
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