Archive for Mathematical Logic

, Volume 46, Issue 5–6, pp 457–464 | Cite as

Antichains in partially ordered sets of singular cofinality

  • Assaf Rinot


In their paper from 1981, Milner and Sauer conjectured that for any poset \(\langle P,\le\rangle\), if \(cf(P,\le)=\lambda>cf(\lambda)=\kappa\), then P must contain an antichain of size κ. We prove that for λ > cf(λ) = κ, if there exists a cardinal μ < λ such that cov(λ, μ, κ, 2) = λ, then any poset of cofinality λ contains λ κ antichains of size κ. The hypothesis of our theorem is very weak and is a consequence of many well-known axioms such as GCH, SSH and PFA. The consistency of the negation of this hypothesis is unknown.


Poset Antichain Singular cofinality 

Mathematics Subject Classification (2000)

03E04 03E35 06A07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fraïssé, R.: Theory of relations, revised edn, with an appendix by Norbert Sauer. Stud. Logic Found. Math. North-Holland, Amsterdam, ii + 451 pp. (2000)Google Scholar
  2. 2.
    Gorelic I. (2006). External cofinalities and the antichain condition in partial orders. Ann. Pure Appl. Logic 140(1–3): 104–109 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hajnal A. and Milner E.C. (1991). On k-independent subsets of a closure. Stud. Sci. Math. Hung. 26(4): 467–470 zbMATHMathSciNetGoogle Scholar
  4. 4.
    Hajnal A. and Sauer N. (1986). Complete subgraphs of infinite multipartite graphs and antichains in partially ordered sets. Discrete Math. 59(1–2): 61–67 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Magidor, M.: unpublished notes (2002)Google Scholar
  6. 6.
    Milner, E.C.: Recent results on the cofinality of ordered sets. In: (L’Arbresle 1982) Orders: description and roles, vol. 99 of North-Holland Math. Stud. pp. 1–8. North-Holland, (1984)Google Scholar
  7. 7.
    Milner, E.C., Pouzet, M.: On the cofinality of partially ordered sets. In: (Banff, Alta. 1981) Ordered sets, vol. 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., pp. 279–298. Reidel, Dordrecht (1982)Google Scholar
  8. 8.
    Milner E.C. and Pouzet M. (1985). On the independent subsets of a closure system with singular dimension. Algebra Universalist 21(1): 25–32 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Milner, E.C., Pouzet, M.: Posets with singular cofinality. Preprint (1997)Google Scholar
  10. 10.
    Milner E.C. and Prikry K. (1983). The cofinality of a partially ordered set. Proc. Lond. Math. Soc. 46(3): 454–470 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Milner E.C. and Sauer N. (1981). Remarks on the cofinality of a partially ordered set and a generalization of König’s lemma. Discrete Math. 35: 165–171 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pouzet, M.: Parties cofinales des ordres partiels ne contenant pas d’antichaines infinies. Unpublished paper (1980)Google Scholar
  13. 13.
    Rinot, A.: Aspects of singular cofinality. Preprint (2006)Google Scholar
  14. 14.
    Rinot A. (2006). On the consistency strength of the Milner–Sauer conjecture. Ann. Pure Appl. Logic 140(1–3): 110–119 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Shelah, S.: Cardinal arithmetic, vol. 29 of Oxford Logic Guides. The Clarendon Press, Oxford University Press, Oxford Science Publications,Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations