Erdős–Ko–Rado theorem

The set of Erdős -Ko - Rado is a set of set theory. It is named after its authors Paul Erdős, Chao Ko and Richard Rado. The set is an upper limit on the cardinality of a k- intersecting family (k- uniform intersecting family) in an n-set for.

Statement

The cardinality of a k- intersecting family in an n-set is limited by for.

Comments

A k- average family, applies to the equality, the set of all k-sets that include a fixed element of the n-element set.

A simple proof is G.O.H. Katona in the Journal of Combinatorial Theory (B). This proof is by double counting. The original proof of 1961 used induction on n

The case is trivial, because then each have two k-sets a non- empty intersection and are obtained.

Paul Erdős, Rado Richard Chao Ko and published the sentence in 1961, but he was already formulated in 1938 during the joint stay of the authors at Cambridge. The ground for such a long time difference Erdos is the lack of interest in combinatorics in that time.

Swell

  • Martin Aigner, Günter M. Ziegler: Proofs from THE BOOK, Springer, Berlin 2002, ISBN 3-540-42535-7 ( 3rd edition: ISBN 978-3-642-02258-6 )
  • Paul Erdős: My joint work with Richard Rado, Surveys of Combinatorics, Cambridge, 1987, p.53 -80
  • Stasys Junka: Extremal Combinatorics, Springer, Berlin 2001, ISBN 3-540-66313-4
  • Paul Erdős, Richard Rado, Chao Ko: Intersection theorems for systems of finite sets, Quarterly Journal of Mathematics, Oxford Series ( 1961), series 2 12: 313-320.
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