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The maximum number of disjoint pairs in a family of subsets

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Abstract

Let ℱ be a family of 2n+1 subsets of a 2n-element set. Then the number of disjoint pairs in ℱ is bounded by (1+o(1))22n. This proves an old conjecture of Erdös. Let ℱ be a family of 21/(k+1)+δ)n subsets of ann-element set. Then the number of containments in ℱ is bounded by (1-1/k+o(1))( |ℱ|2 ). This verifies a conjecture of Daykin and Erdös. A similar Erdös-Stone type result is proved for the maximum number of disjoint pairs in a family of subsets.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    N. Alon

  2. U.E.R. Mathématiques, Université de Paris, VII, 75005, Paris, France

    P. Frankl

Authors
  1. N. Alon
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  2. P. Frankl
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Research supported in part by the Weizmann Fellowship for Scientific Research.

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Alon, N., Frankl, P. The maximum number of disjoint pairs in a family of subsets. Graphs and Combinatorics 1, 13–21 (1985). https://doi.org/10.1007/BF02582924

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  • DOI: https://doi.org/10.1007/BF02582924

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