TY - JOUR
T1 - Lower bounds on coloring numbers from hardness hypotheses in PCF theory
AU - Shelah, Saharon
N1 - Publisher Copyright:
© 2016 American Mathematical Society.
PY - 2016
Y1 - 2016
N2 - We prove that the statement “for every infinite cardinal κ, every graph with list-chromatic number κ has coloring number at most ℶω (κ)” 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 ℶω (κ) 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.
AB - We prove that the statement “for every infinite cardinal κ, every graph with list-chromatic number κ has coloring number at most ℶω (κ)” 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 ℶω (κ) 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.
UR - http://www.scopus.com/inward/record.url?scp=84992397375&partnerID=8YFLogxK
U2 - 10.1090/proc/13163
DO - 10.1090/proc/13163
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AN - SCOPUS:84992397375
SN - 0002-9939
VL - 144
SP - 5371
EP - 5383
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 12
ER -