Antichains in partially ordered sets of singular cofinality

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In their paper from 1981, Milner and Sauer conjectured that for any poset 〈P,≤〉, if cf〈P,≤〉=λ>cfλ=κ, then P must contain an antichain of size κ. We prove that for λ > cf(λ)=κ, if there exists a cardinalμ<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.

Original languageEnglish
Pages (from-to)457-464
Number of pages8
JournalArchive for Mathematical Logic
Issue number5-6
StatePublished - Jul 2007
Externally publishedYes


  • Antichain
  • Poset
  • Singular cofinality


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