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.
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Rinot, A. Antichains in partially ordered sets of singular cofinality. Arch. Math. Logic 46, 457–464 (2007). https://doi.org/10.1007/s00153-007-0049-z
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DOI: https://doi.org/10.1007/s00153-007-0049-z