Cross-cuts in the power set of an infinite set |
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| Authors: | J E Baumgartner P Erdös D Higgs |
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| Institution: | (1) Department of Mathematics, Dartmouth College, 03755 Hanover, NH, USA;(2) Mathematics Institute, Hungarian Academy of Sciences, 1053 Budapest V, Hungary;(3) Department of Pure Mathematics, University of Waterloo, N2L 3G1 Waterloo, Ontario, Canada |
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| 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 satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE= , 1, and 2, 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 and 1.The work reported here has been partially supported by NSERC Grant No. A8054. |
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| Keywords: | primary 04A20 secondary 06A10 04A30 |
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