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Scales with various kinds of good points. (English) Zbl 1521.03134

Summary: We study variants of (more or less) known square-like principles obtained by replacing equality with covering. We show that these weaker principles are still strong enough to yield scales with many good (or even better) points. Such scales can be used to construct large pseudo-Kurepa families.

MSC:

03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
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[1] J.Cummings, M.Foreman, and M.Magidor, Squares, scales and stationary reflection, J. Math. Log.1, 35-98 (2001). · Zbl 0988.03075
[2] H. D.Donder and J. P.Levinski, Stationary sets and game filters, unpublished (appr. 1987).
[3] M.Džamonja and S.Shelah, On squares, outside guessing of clubs and \(I_{< f} [ \lambda ]\), Fund. Math.148, 165-198 (1995). · Zbl 0839.03031
[4] T.Eisworth, On ideals related to \(I [ \lambda ]\), Notre Dame J. Form. Log.46, 301-307 (2005). · Zbl 1092.03021
[5] T.Eisworth, Successors of singular cardinals, in: Handbook of Set Theory, Volume 2, edited by M.Foreman (ed.) and A.Kanamori (ed.) (Springer, 2010), pp. 1229-1350. · Zbl 1198.03049
[6] M.Foreman and M.Magidor, A very weak square principle, J. Symb. Log.62, 175-196 (1997). · Zbl 0880.03022
[7] M.Holz, K.Steffens, and E.Weitz, Introduction to Cardinal Arithmetic, Birkhäuser Advanced Texts: Basler Lehrbücher (Birkhäuser, 1999). · Zbl 0930.03053
[8] T.Huuskonen, T.Hyttinen, and M.Rautila, On the κ‐cub game on λ and \(I [ \lambda ]\), Arch. Math. Log.38, 549-557 (1999). · Zbl 0959.03030
[9] C.Lambie‐Hanson and M.Magidor, On the strength and weaknesses of weak squares, in: Appalachian Set Theory 2006-2012, edited by J.Cummings (ed.) and E.Schimmerling (ed.), London Mathematical Society Lecture Series Vol. 406 (Cambridge University Press, 2013), pp. 301-330. · Zbl 1367.03079
[10] P.Matet, Large cardinals and covering numbers, Fund. Math.205, 45-75 (2009). · Zbl 1191.03032
[11] P.Matet, Weak saturation of ideals on \(\wp_\kappa ( \lambda )\), Math. Log. Q.57, 149-165 (2011). · Zbl 1237.03030
[12] P.Matet, Normal restrictions of the non‐cofinal ideal on \(\wp_\kappa ( \lambda )\), Fund. Math.221, 1-22 (2013). · Zbl 1300.03023
[13] P.Matet, Ideals on \(\wp_\kappa ( \lambda )\) associated with games of uncountable length, Arch. Math. Log.54, 291-328 (2015). · Zbl 1375.03055
[14] P.Matet, Versions for \(\wp_\kappa ( \lambda )\) of two results of Shelah on diamond, Part I, preprint.
[15] A.Rinot, A relative of the approachability ideal, diamond and non‐saturation, J. Symb. Log.75, 1035-1065 (2010). · Zbl 1203.03074
[16] S.Shelah, Reflecting stationary sets and successors of singular cardinals, Arch. Math. Log.31, 25-53 (1991). · Zbl 0742.03017
[17] S.Shelah, Cardinal Arithmetic, Oxford Logic Guides Vol. 29 (Oxford University Press, 1994). · Zbl 0848.03025
[18] S.Shelah, The Generalized Continuum Hypothesis revisited, Isr. J. Math.116, 285-321 (2000). · Zbl 0955.03054
[19] S.Todorčević, Partitioning pairs of countable sets, Proc. Amer. Math. Soc.111, 841-844 (1991). · Zbl 0722.03036
[20] S.Todorčević, Kurepa families and cofinal similarities, handwritten notes (1989).
[21] S.Todorčević, Cofinal Kurepa families, handwritten notes (1990).
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