Lower bounds on coloring numbers from hardness hypotheses in PCF theory

TL;DR

This paper explores the relationship between coloring number bounds in infinite graphs and set-theoretic hypotheses, showing that improving these bounds relates to deep open problems in cardinal arithmetic.

Contribution

It establishes that the optimality of Kojman's upper bound on coloring numbers is equivalent to a fundamental open problem in cardinal arithmetic, linking graph theory and set theory.

Findings

Proves that certain coloring bounds imply the RGCH theorem.

Shows that improving bounds relates to open problems in cardinal arithmetic.

Highlights the independence of these bounds from ZFC set theory.

Abstract

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.