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Incompatible Intersection Properties

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Abstract

Let F ⊂ 2 [n] be a family in which any three sets have non-empty intersection and any two sets have at least 32 elements in common. The nearly best possible bound F ≤ 2n−2 is proved. We believe that 32 can be replaced by 3 and provide a simple-looking conjecture that would imply this.

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Acknowledgments

We thank Stijn Cambie for his remarks, presented in the previous section, and useful comments on the presentation of the paper. We also thank the anonymous referees for their helpful comments.

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Authors and Affiliations

  1. Rényi Institute, Budapest, Hungary

    Peter Frankl

  2. University of Oxford, Oxford, UK

    Andrey Kupavskii

  3. Moscow Institute of Physics and Technology, Moscow, Russia

    Andrey Kupavskii

  4. Caucasus Mathematical Center, Adyghe State University, Maykop, Russia

    Andrey Kupavskii

Authors
  1. Peter Frankl
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  2. Andrey Kupavskii
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Corresponding author

Correspondence to Andrey Kupavskii.

Additional information

The research of the second author was supported by the EPSRC grant no. EP/N019504/1, the Russian Foundation for Basic Research (grant no. 18-01-00355) and the Council for the Support of Leading Scientific Schools of the President of the Russian Federation (grant no. N.Sh.-6760.2018.1).

A property is simply a class of families, and a family has that property if and only if it belongs to the class.

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Frankl, P., Kupavskii, A. Incompatible Intersection Properties. Combinatorica 39, 1255–1266 (2019). https://doi.org/10.1007/s00493-019-4064-6

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  • DOI: https://doi.org/10.1007/s00493-019-4064-6

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