Intersections ofk-element sets

Abstract

LetF be a collection ofk-element sets with the property that the intersection of no two should be included in a third. We show that such a collection of maximum size satisfies .2715k+o(k)≦≦log2 |F|≦.7549k+o(k) settling a question raised by Erdős. The lower bound is probabilistic, the upper bound is deduced via an entropy argument. Some open questions are posed.

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References

  1. [1]

    P. Erdős, private communication.

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    R. McEliece,The Theory of Information and Coding, Encyclopedia of Mathematics, Vol. 3, Addison-Wesley, Reading, Massachusetts, 1977.

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This research has been supported in part by the Office of Naval Research under Contract N00014-76-C-0366.

Supported in part by a NSF postdoctoral Fellowship.

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Kleitman, D.J., Shearer, J. & Sturtevant, D. Intersections ofk-element sets. Combinatorica 1, 381–384 (1981). https://doi.org/10.1007/BF02579460

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AMS subject classification (1980)

  • 05 C 65
  • 05 C 35