Complete subgraphs of infinite multipartite graphs and antichains in partially ordered sets
The authors investigate a conjecture of E. C. Milner and Sauer: ``If the cofinality of a partially ordered set (P,\(\leq)\) is a singular cardinal \(\lambda\), then P contains an antichain of size cf(\(\lambda)\)''. (The cofinality of (P,\(\leq)\) is the least cardinality of a cofinal subset. The word ``antichain'' has the nonstandard meaning of a set of pairwise incomparable elements.) The conjecture was inspired by a result of \textit{M. Pouzet} [``Parties cofinals des ordres partiels ne contenant pas d'antichaines infinies'', J. Lond. Math. Soc., 2nd Ser. (to appear)] concluding under the same hypothesis that P contains an infinite antichain. The authors prove the conjecture under certain additional assumptions on the cardinal \(\lambda\). They also prove that, if the conjecture fails for some cardinal \(\lambda\), then there must exist some \(\lambda '<\lambda\) and a set S of subsets of \(\lambda\) ', ordered by inclusion, which provides a counterexample.
scientific article; zbMATH DE number 4068875