Abstract
In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y ⊑E satisfyX⊑Y, X ⊑ someX′ inC, andY⊑ someY′ inC, thenX⊑Z⊑Y for someZ inC. ForE=ω, ω1, and ω2, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for ω and ω1.
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References
Recent Progress in Combinatorics, Proceedings of the Third Waterloo Conference on Combinatorics, May 1968, Academic Press, New York, 1969, pp. 343–344.
D. Higgs (1969) Equicardinality of bases in B-matroids,Canad. Math. Bull. 12, 861–862.
W. Sierpiński (1956)Hypothèse du Continu, 2nd edn., Chelsea, New York.
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Communicated by F. Galvin
The work reported here has been partially supported by NSERC Grant No. A8054.
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Baumgartner, J.E., Erdös, P. & Higgs, D. Cross-cuts in the power set of an infinite set. Order 1, 139–145 (1984). https://doi.org/10.1007/BF00565649
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DOI: https://doi.org/10.1007/BF00565649