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Antichains in partially ordered sets of singular cofinality. (English) Zbl 1115.03057

Summary: In their paper from 1981, E. C. Milner and N. Sauer [Discrete Math. 35, 165–171 (1981; Zbl 0465.05041)] conjectured that for any poset \(\langle P,\leq\rangle\), if \(\text{cf}(P,\leq)= \lambda> \text{cf}(\lambda)=\kappa\), then \(P\) must contain an antichain of size \(\kappa\). We prove that for \(\lambda >\) cf\((\lambda) = \kappa\), if there exists a cardinal \(\mu < \lambda\) such that cov\((\lambda, \mu, \kappa, 2) = \lambda\), then any poset of cofinality \(\lambda\) contains \(\lambda^{\kappa}\) antichains of size \(\kappa\). 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.

MSC:

03E04 Ordered sets and their cofinalities; pcf theory
03E35 Consistency and independence results
06A07 Combinatorics of partially ordered sets

Citations:

Zbl 0465.05041
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References:

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