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Lower bounds on coloring numbers from hardness hypotheses in pcf theory


Author: Saharon Shelah
Journal: Proc. Amer. Math. Soc. 144 (2016), 5371-5383
MSC (2010): Primary 03E04; Secondary 03E05, 03C15
DOI: https://doi.org/10.1090/proc/13163
Published electronically: July 28, 2016
MathSciNet review: 3556279
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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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Additional Information

Saharon Shelah
Affiliation: Department of Mathematics, Hebrew University of Jerusalem, Jerusalem, 9190401 Israel

DOI: https://doi.org/10.1090/proc/13163
Received by editor(s): December 13, 2014
Received by editor(s) in revised form: February 22, 2016
Published electronically: July 28, 2016
Additional Notes: The author thanks the Israel Science Foundation for partial support of this research, Grant no. 1053/11. Publication 1052
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2016 American Mathematical Society

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