Abstract
We study the Galvin property. We show that various square principles imply that the cofinality of the Galvin number is uncountable (or even greater than \(\aleph _1\)). We prove that the proper forcing axiom is consistent with a strong negation of the Glavin property.
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References
Abraham, U., Magidor, M. Cardinal arithmetic, Handbook of set theory, vols. 1, 2, 3, (Springer, Dordrecht, 2010, pp. 1149–1227)
Baumgartner, J. E., Ha̧jņal, A. Mate, A. Weak saturation properties of ideals, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), vol. I, North-Holland, Amsterdam, 1975, pp. 137–158. Colloq. Math. Soc. János Bolyai, Vol. 10
Baumgartner, J. E. Applications of the proper forcing axiom, Handbook of set-theoretic topology, (North-Holland, Amsterdam, 1984, pp. 913–959)
Blass, A. Combinatorial cardinal characteristics of the continuum, Handbook of set theory. vols. 1, 2, 3, (Springer, Dordrecht, 2010, pp. 395–489)
J. Cummings, Notes on singular cardinal combinatorics. Notre Dame J. Formal Logic 4 6(3), 251–282 (2005)
Cummings, J. Iterated forcing and elementary embeddings, Handbook of set theory. vols. 1, 2, 3 (Springer, Dordrecht, 2010, pp. 775–883)
J. Cummings, M. Foreman, M. Magidor, Squares, scales and stationary reflection. J. Math. Log. 1(1), 35–98 (2001)
J. Cummings, M. Magidor, Martin’s maximum and weak square. Proc. Am. Math. Soc. 139(9), 3339–3348 (2011)
Garti S. Weak diamond and Galvin’s property. Period. Math. Hungar. 74(1), 128–136 (2017)
S. Garti, Tiltan. C. R. Math. Acad. Sci. Paris 356(4), 351–359 (2018)
A. Sharon, M. Viale, Some consequences of reflection on the approachability ideal. Trans. Am. Math. Soc. 362(8), 4201–4212 (2010)
S. Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29 (The Clarendon Press, Oxford University Press, Oxford Science Publications, New York, 1994)
Acknowledgements
We are deeply indebted to the referee of the paper. The work of the referee on our paper included many mathematical corrections, new suggestions (some of which are integrated within the manuscript) and a considerable improvement of the exposition.
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Garti, S., Hayut, Y., Horowitz, H. et al. Forcing axioms and the Galvin number. Period Math Hung 84, 250–258 (2022). https://doi.org/10.1007/s10998-021-00407-9
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DOI: https://doi.org/10.1007/s10998-021-00407-9