# Sh:1052

- Shelah, S. (2016).
*Lower bounds on coloring numbers from hardness hypotheses in pcf theory*. Proc. Amer. Math. Soc.,**144**(12), 5371–5383. arXiv: 1503.02423 DOI: 10.1090/proc/13163 MR: 3556279 -
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 [koj] using the RGCH theorem [sh:460] implies the RGCG theorem via a short forcing argument. By the same method, a better upper bound than \beth_\omega(\nu) in this statement implies stronger forms of the RGCH theorem whose consistency as well as the consistency of their negations are wide open. Thus, the optimality of Kojman’s upper bound is a purely cardinal arithmetic problem, which, as discussed below, may be quite hard to decide. - Version 2016-02-18_11 (14p) published version (13p)

Bib entry

@article{Sh:1052, author = {Shelah, Saharon}, title = {{Lower bounds on coloring numbers from hardness hypotheses in pcf theory}}, journal = {Proc. Amer. Math. Soc.}, fjournal = {Proceedings of the American Mathematical Society}, volume = {144}, number = {12}, year = {2016}, pages = {5371--5383}, issn = {0002-9939}, mrnumber = {3556279}, mrclass = {03E04 (03C15 03E05)}, doi = {10.1090/proc/13163}, note = {\href{https://arxiv.org/abs/1503.02423}{arXiv: 1503.02423}}, arxiv_number = {1503.02423} }