Combinatorica

, Volume 4, Issue 2, pp 247–257

The exact bound in the Erdös-Ko-Rado theorem

Authors

  • Richard M. Wilson
    • Department of MathematicsCalifornia Institute of Technology
Article

DOI: 10.1007/BF02579226

Cite this article as:
Wilson, R.M. Combinatorica (1984) 4: 247. doi:10.1007/BF02579226

Abstract

This paper contains a proof of the following result: ifn≧(t+1)(kt−1), then any family ofk-subsets of ann-set with the property that any two of the subsets meet in at leastt points contains at most\(\left( {\begin{array}{*{20}c} {n - t} \\ {k - t} \\ \end{array} } \right)\) subsets. (By a theorem of P. Frankl, this was known whent≧15.) The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdös, Ko, and Rado which reaches the same conclusion under the hypothesisnt+(kt)\(\left( {\begin{array}{*{20}c} k \\ t \\ \end{array} } \right)^3 \). Our proof is linear algebraic in nature; it may be considered as an application of Delsarte’s linear programming bound, but somewhat lengthy calculations are required to reach the stated result. (A. Schrijver has previously noticed the relevance of these methods.) Our exposition is self-contained.

AMS subject classification (1980)

05 C 3505 C 65
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Copyright information

© Akadémiai Kiadó 1984