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On incomparable collections of sets

  • P. D. Seymour (a1)

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A. J. W. Hilton [5] conjectured that if P, Q are collections of subsets of a finite set S, with |S| = n, and |P| > 2n−2, |Q| ≥ 2n−2, then for some AP, BQ we have AB or BA. We here show that this assertion, indeed a stronger one, can be deduced from a result of D. J. Kleitman. We then give another proof of a recent result also proved by Lovász and by Schönheim.

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1. Daykin, D. E. and Lovasz, L., “The number of values of Boolean functions” (submitted).
2. Edmonds, J. and Fulkerson, D. R., “Bottleneck Extrema”, J. Combinatorial Theory, 8 (1970), 299306.
3. Kleitman, D. J., “Families of non-disjoint subsets”, J. Combinatorial Theory, 1 (1966), 153155.
4. Schönheim, J., “Ideals of sets ”, Proc. British Combinatorial Conference Aberystwyth (1973) (forthcoming).
5. Brace, A. and Daykin, D. E., “Spemer type theorems for finite sets ”, Combinatorics (Proc. Combinatorial Conference, Oxford, 1972).
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