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On intersecting families of finite sets

Published online by Cambridge University Press:  17 April 2009

Peter Frankl
Peter Frankl
Affiliation:
CNRS, 7 Quai Anatole France, 75007 Paris, France.
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Let F be a family of k-element subsets of an n-set, n > n0(k). Suppose any two members of F have non-empty intersection. Let τ(F) denote min|T|, T meets every member of F. Erdös, Ko and Rado proved and that if equality holds then τ(F) = 1. Hilton and Milner determined max|F| for τ(F) = 2. In this paper we solve the problem for τ(F) = 3.

The extremal families look quite complicated which shows the power of the methods used for their determination.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 21 , Issue 3 , June 1980 , pp. 363 - 372
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Erdös, P. and Rado, R., “Intersection theorems for systems of sets”, J. London Math. Soc. 35 (1960), 85–90.CrossRefGoogle Scholar
[2]Erdös, P., Ko, Chao, and Rado, R., “Intersection theorems for systems of finite sets”, Quart. J. Math. Oxford (2) 12 (1961), 313–320.CrossRefGoogle Scholar
[3]Frankl, Peter, “On families of finite sets no two of which intersect in a singleton”, Bull. Austral. Math. Soc. 17 (1977), 125–134.CrossRefGoogle Scholar
[4]Hilton, A.J.W. and Milner, E.C., “Some intersection theorems for systems of finite sets”, Quart. J. Math. Oxford (2) 18 (1967), 369–384.CrossRefGoogle Scholar