Abstract
We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \(\mathsf {cov}(\mathcal {M})\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, pp. 395–489. Springer, Berlin (2010)
Goldstern, M., Repický, M., Shelah, S., Spinas, O.: On tree ideals. Proc. Am. Math. Soc. 123(5), 1573–1581 (1995)
Jech, T.: Set Theory: The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin (2006)
Judah, H., Miller, A.W., Shelah, S.: Sacks forcing, Laver forcing, and Martin’s axiom. Arch. Math. Logic 31(3), 145–161 (1992)
Jossen, S., Spinas, O.: A two-dimensional tree ideal. In: Logic Colloquium 2000. Lecture Notes in Logic 19, pp. 294–322. ALS (2005)
Kechris, A.S.: On a notion of smallness for subsets of the Baire space. Trans. Am. Math. Soc. 229, 191–207 (1977)
Kunen, K.: Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)
Monk, J.D., Koppelberg, S., Bonnet, R.: Handbook of Boolean Algebras: General Theory of Boolean Algebras. North-Holland, Amsterdam (1989)
Rosłanowski, A., Shelah, S.: More forcing notions imply diamond. Arch. Math. Logic 35(5), 299–313 (1996)
Saint-Raymond, J.: Approximation des sous-ensembles analytiques par l’intérieur. C. R. Acad. Sci. Paris Sér. AB 281, A85–A87 (1975)
Shelah, S.: Vive la différence I: nonisomorphism of ultrapowers of countable models. In: Judah, H., Just, W., Woodin, H. (eds.) Set Theory of the Continuum, pp. 357–405. Springer, Berlin (1992)
Simon, P.: Sacks forcing collapses c to b. Comment. Math. Univ. Carol. 34(4), 707–710 (1993)
Spinas, O.: Analytic countably splitting families. J. Symb. Logic 69, 101–117 (2004)
Spinas, O.: Splitting squares. Isr. J. Math. 162(1), 57–73 (2007)
Spinas, O.: Silver trees and Cohen reals. Isr. J. Math. 211(1), 473–480 (2016)
Spinas, O.: Generic trees. J. Symb. Logic 60(3), 705–726 (1995)
Spinas, O.: Partition numbers. Ann. Pure Appl. Logic 90(1), 243–262 (1997)
Shelah, S., Spinas, O.: Different cofinalities of tree ideals. Submitted (2017)
Spinas, O., Wyszkowski, M.: Silver antichains. J. Symb. Logic 80(2), 503–519 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors would like to thank the DFG for partial support (Grant SP683/4-1).
Rights and permissions
About this article
Cite this article
Hein, P., Spinas, O. Antichains of perfect and splitting trees. Arch. Math. Logic 59, 367–388 (2020). https://doi.org/10.1007/s00153-019-00694-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-019-00694-7