Lower bounds on coloring numbers from hardness hypotheses in pcf theory

Shelah, Saharon

doi:10.1090/proc/13163

Lower bounds on coloring numbers from hardness hypotheses in pcf theory
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by Saharon Shelah

Proc. Amer. Math. Soc. 144 (2016), 5371-5383

DOI: https://doi.org/10.1090/proc/13163

Published electronically: July 28, 2016

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Abstract:

We prove that the statement “for every infinite cardinal $\kappa$, every graph with list-chromatic number $\kappa$ has coloring number at most $\beth _\omega (\kappa )$” 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 $\beth _\omega (\kappa )$ 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.

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