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Archive for Mathematical Logic

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

Antichains in partially ordered sets of singular cofinality

  • Assaf Rinot
Article
  • 37 Downloads

Abstract

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.

Keywords

Poset Antichain Singular cofinality 

Mathematics Subject Classification (2000)

03E04 03E35 06A07 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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