Abstract
We consider several game versions of the cardinal invariants \({\mathfrak {t}}\), \({\mathfrak {u}}\) and \({\mathfrak {a}}\). We show that the standard proof that parametrized diamond principles prove that the cardinal invariants are small actually shows that their game counterparts are small. On the other hand we show that \({\mathfrak {t}}<{\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}<{\mathfrak {u}}_{Builder}\) are both relatively consistent with ZFC, where \({\mathfrak {t}}_{Builder}\) and \({\mathfrak {u}}_{Builder}\) are the principal game versions of \({\mathfrak {t}}\) and \({\mathfrak {u}}\), respectively. The corresponding question for \({\mathfrak {a}}\) remains open.
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References
Balcar, B., Doucha, M., Hrušák, M.: Base tree property. Order 32(1), 69–81 (2015)
Balcar, B., Pelant, J., Simon, P.: The space of ultrafilters on \({\mathbb{N}}\) covered by nowhere dense sets. Fund. Math. 110, 11–24 (1980)
Bartoszyński, T., Judah, H.: Set Theory. On the Structure of the Real Line. A K Peters, Wellesley (1995)
Baumgartner, J.E., Dordal, P.: Adjoining dominating functions. J. Symb. Log. 50(1), 94–101 (1985)
Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set theory, pp. 395–489. Springer, Berlin (2009)
Brendle, J., Raghavan, D.: Bounding, splitting, and almost disjointness. Ann. Pure Appl. Log. 165(2), 631–651 (2014)
Brendle, J., Shelah, S.: Ultrafilters on \(\omega \)—their ideals and their cardinal characteristics. Trans. Am. Math. Soc. 351(7), 2643–2674 (1999)
Devlin, K.J., Shelah, S.: A weak version of \(\diamondsuit \) which follows from \(2^{\aleph _0}<2^{\aleph _1}\). Israel J. Math. 29(2), 239–247 (1978)
Dow, A.: More set-theory for topologists. Topol. Appl. 64(3), 243–300 (1995)
Foreman, M.: Games played on Boolean algebras. J. Symb. Log. 48(3), 714–723 (1983)
Jensen, R.B.: The fine structure of the constructible hierarchy. Ann. Math. Log. 4, 229–308 (1972)
Judah, H., Shelah, S.: \(\varvec {\Delta }^1_2\)-sets of reals. Ann. Pure Appl. Log. 42, 207–223 (1989)
Judah, H., Shelah, S.: Q-sets, Sierpinski sets, and rapid filters. Proc. Am. Math. Soc. 111, 821–832 (1991)
Malliaris, M., Shelah, S.: Cofinality spectrum theorems in model theory, set theory, and general topology. J. Am. Math. Soc. 29(1), 237–297 (2016)
Moore, J.T., Hrušák, M., Džamonja, M.: Parametrized \(\diamondsuit \)-principles. Trans. Am. Math. Soc. 356(6), 2281–2306 (2004)
Shelah, S.: Proper and Improper Forcing, Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)
Vojtáš, P.: Game properties of Boolean algebras. Comment. Math. Univ. Carol. 24(2), 349–369 (1983)
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The first-listed author was supported by Grants-in-Aid for Scientific Research (C) 15K04977 and 18K03398, Japan Society for the Promotion of Science. The second-listed author was supported by a PAPIIT Grants IN 102311 and IN 100317, and a CONACyT Grant 177758. The third-listed author was supported by the Austrian Science Fund (FWF) under the Projects P 26869-N25 and P 29860-N35.
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Brendle, J., Hrušák, M. & Torres-Pérez, V. Construction with opposition: cardinal invariants and games. Arch. Math. Logic 58, 943–963 (2019). https://doi.org/10.1007/s00153-019-00671-0
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DOI: https://doi.org/10.1007/s00153-019-00671-0