Lambie-Hanson, Chris; Rinot, Assaf Knaster and friends II: the C-sequence number. (English) Zbl 07355297 J. Math. Log. 21, No. 1, Article ID 2150002, 54 p. (2021). Summary: Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the \(C\)-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of \(\mathsf{ZFC}\) and independence results about the \(C\)-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general \(C\)-sequence spectrum and uncover some tight connections between the \(C\)-sequence spectrum and the strong coloring principle \(\mathrm{U}(\dots)\), introduced in Part I of this series. Cited in 8 Documents MSC: 03E35 Consistency and independence results 03E05 Other combinatorial set theory 03E55 Large cardinals 06E10 Chain conditions, complete algebras Keywords:C-sequence number; closed coloring; unbounded functions; indexed square PDFBibTeX XMLCite \textit{C. Lambie-Hanson} and \textit{A. Rinot}, J. Math. Log. 21, No. 1, Article ID 2150002, 54 p. (2021; Zbl 07355297) Full Text: DOI arXiv References: [1] Brodsky, A. M. and Rinot, A., Distributive Aronszajn trees, Fund. Math.245(3) (2019) 217-291. · Zbl 1472.03043 [2] Cox, S. and Lücke, P., Characterizing large cardinals in terms of layered posets, Ann. Pure Appl. Logic168(5) (2017) 1112-1131. · Zbl 1422.03094 [3] Cummings, J., Foreman, M. and Magidor, M., Squares, scales and stationary reflection, J. Math. Log.1(1) (2001) 35-98. · Zbl 0988.03075 [4] Cummings, J. and Magidor, M., Martin’s maximum and weak square, Proc. Amer. Math. Soc.139(9) (2011) 3339-3348. · Zbl 1270.03098 [5] Gitik, M., Prikry-type forcings, in Handbook of Set Theory, Vols. 1-3 (Springer, Dordrecht, 2010), pp. 1351-1447. · Zbl 1198.03062 [6] Gitik, M. and Sharon, A., On SCH and the approachability property, Proc. Amer. Math.Soc.136(1) (2008) 311-320. · Zbl 1140.03033 [7] Hayut, Y. and Lambie-Hanson, C., Simultaneous stationary reflection and square sequences, J. Math. Log.17(2) (2017), Article ID:1750010, 27pp. · Zbl 1423.03164 [8] Y. Hayut and S. Unger, Stationary reflection (2018), http://arxiv.org/abs/1804. 11329. [9] Jech, T., Stationary subsets of inaccessible cardinals, in Axiomatic Set Theory (Boulder, Colorado., 1983), Vol. 31 (American Mathematical Society, Providence, RI, 1984), pp. 115-142. [10] Jech, T., Set Theory, (Springer-Verlag, Berlin, 2003). The third millennium edition, revised and expanded. [11] Krueger, J. and Schimmerling, E., An equiconsistency result on partial squares, J. Math. Log.11(1) (2011) 29-59. · Zbl 1258.03068 [12] Kunen, K., Saturated ideals, J. Symbolic Logic43(1) (1978) 65-76. · Zbl 0395.03031 [13] Lambie-Hanson, C., Squares and narrow systems, J. Symb. Log.82(3) (2017) 834-859. · Zbl 1422.03107 [14] Lambie-Hanson, C. and Lücke, Philipp, Squares, ascent paths, and chain conditions, J. Symb. Log.83(4) (2018) 1512-1538. · Zbl 1477.03181 [15] Lambie-Hanson, C. and Rinot, A., Knaster and friends I: Closed colorings and precalibers, Algebra Universalis79(4) (2018) Article ID: 90, 39pp. · Zbl 1522.03216 [16] Levine, M. and Rinot, A., Partitioning a reflecting stationary set, Proc. Amer. Math. Soc.148(8) (2020) 3551-3565. · Zbl 1484.03085 [17] Mekler, A. H. and Shelah, S., The consistency strength of “every stationary set reflects”, Israel J. Math.67 (1989) 353-366. · Zbl 0694.03033 [18] Neeman, I. and Steel, J., Equiconsistencies at subcompact cardinals, Arch. Math. Logic55(1) (2016) 207-238. · Zbl 1402.03070 [19] Rinot, A., Chain conditions of products, and weakly compact cardinals, Bull. Symbolic Logic20(3) (2014) 293-314. · Zbl 1345.03092 [20] Rinot, A., Complicated colorings, Math. Res. Lett.21(6) (2014) 1367-1388. · Zbl 1320.03076 [21] Todorcevic, S., Partitioning pairs of countable ordinals, Acta Math.159(3-4) (1987) 261-294. · Zbl 0658.03028 [22] Todorcevic, S., Walks on Ordinals and Their Characteristics, Vol. 263, (Birkhäuser Verlag, Basel, 2007). · Zbl 1148.03004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.