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On union-closed sets and Conway's sequence

Published online by Cambridge University Press:  17 April 2009

J-C. Renaud  and
L.F. Fitina
J-C. Renaud
Affiliation:
Department of Mathematics, University of Papua New Guinea, PO Box 320, University, Papua New Guinea
L.F. Fitina
Affiliation:
Department of Mathematics, University of Papua New Guinea, PO Box 320, University, Papua New Guinea
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Abstract

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In 1991 Renaud defined a boundary function φ(n) for union-closed sets, and evaluated it to n = 17. Also in 1991, Mallows examined a sequence a(n) defined recursively by Conway in 1988.

Investigation of some properties of strictly reduced ordered power sets, a class of union-closed sets, leads to the conclusion that a(n + 1) is an upper bound for φ(n), and the union-closed sets conjecture holds if the conjecture φ(n) = a(n + 1) is valid.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 2 , April 1993 , pp. 321 - 332
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Mallows, C.I., ‘Conway's challenge sequence’, Amer. Math. Monthly 98 (1991), 5–20.CrossRefGoogle Scholar
[2]Renaud, J-C., ‘Is the union-closed sets conjecture the best possible?’, J. Austral. Math. Soc. Ser. A 51 (1991), 276–283.CrossRefGoogle Scholar
[3]Rival, I. (Editor), Graphs and order (Reidel, Dordrecht, 1984).Google Scholar