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On incomparable collections of sets

Part of: Combinatorics

Published online by Cambridge University Press:  26 February 2010

P. D. Seymour
P. D. Seymour
Affiliation:
The Mathematical Institute, Oxford
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Extract

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 A ∈ P, B ∈ Q we have A ⊆ B or B ⊆ A. 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.

MSC classification

Secondary: 05A20: Combinatorial inequalities
Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 208 - 209
Copyright
Copyright © University College London 1973

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References

1. Daykin, D. E. and Lovasz, L., “The number of values of Boolean functions” (submitted).Google Scholar
2. Edmonds, J. and Fulkerson, D. R., “Bottleneck Extrema”, J. Combinatorial Theory, 8 (1970), 299–306.CrossRefGoogle Scholar
3. Kleitman, D. J., “Families of non-disjoint subsets”, J. Combinatorial Theory, 1 (1966), 153–155.CrossRefGoogle Scholar
4. Schönheim, J., “Ideals of sets ”, Proc. British Combinatorial Conference Aberystwyth (1973) (forthcoming).Google Scholar
5. Brace, A. and Daykin, D. E., “Spemer type theorems for finite sets ”, Combinatorics (Proc. Combinatorial Conference, Oxford, 1972).Google Scholar