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)

@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}
}