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Families of finite sets satisfying an intersection condition

Published online by Cambridge University Press:  17 April 2009

Peter Frankl
Peter Frankl
Affiliation:
Department of Algebra and Number Theory, Eötvös Lorand University Budapest, Budapest, Hungary.
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Abstract

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The following theorem is proved.

Let X be a finite set of cardinality n ≥ 2, and let F be a family of subsets of X. Suppose that for F1, F2, F3 ∈ F we have |F1 ∩ F2 ∩ F3| ≥ 2. Then |F| ≤ 2n−2with equality holding if and only if for two different elements x, y of X, F = {F ⊆ X | x ∈ F, y ∈ F}.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 1 , August 1976 , pp. 73 - 79
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Erdös, P., Ko, Chao, and Rado, R., “Intersection theorems for systems of finite sets”, Quart. J. Math. Oxford Ser. 12 (1961), 313–320.CrossRefGoogle Scholar
[2]Feller, William, An introduction to probability theory and its applications, Volume II (John Wiley & Sons, New York, London, Sydney, Toronto, 1966; second edition, 1971).Google Scholar
[3]Frankl, Peter, “Families of finite sets satisfying a union-condition”, submitted.Google Scholar
[4]Katona, Gy., “Intersection theorems for systems of finite sets”, Acta Math. Acad. Sci. Hungar. 15 (1964), 329–337.CrossRefGoogle Scholar