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Complete subgraphs of infinite multipartite graphs and antichains in partially ordered sets. (English) Zbl 0655.05002

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 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.

MSC:

05A05 Permutations, words, matrices
03E10 Ordinal and cardinal numbers
06A06 Partial orders, general
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References:

[1] M. Pouzet, Parties cofinals des ordres partiels ne contenant pas d’antichaines infinies, J. London Math. Soc., to appear.; M. Pouzet, Parties cofinals des ordres partiels ne contenant pas d’antichaines infinies, J. London Math. Soc., to appear.
[2] Milner, E. C.; Prikry, K., The cofinality of a partially ordered set, University of Calgary, Research Paper No. 474 (1980), (submitted to London Math. Soc.) · Zbl 0511.06002
[3] Erdös, P.; Hajnal, A.; Rado, R., Partition relations for cardinal numbers, Acta Math. Sci. Hung, 16, 93-196 (1965) · Zbl 0158.26603
[4] S. Shelah, Notes on partition calculus, infinite and finite sets, Colloquia Math. Soc. J. Bolyai.; S. Shelah, Notes on partition calculus, infinite and finite sets, Colloquia Math. Soc. J. Bolyai. · Zbl 0325.04005
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