Advertisement

Classification of Maximum Hittings by Large Families

  • Candida Bowtell
  • Richard MycroftEmail author
Original Paper
  • 3 Downloads

Abstract

For integers r and n, where n is sufficiently large, and for every set \(X \subseteq [n]\) we determine the maximal left-compressed intersecting families \({\mathcal {A}}\subseteq \left( {\begin{array}{c}[n]\\ r\end{array}}\right) \) which achieve maximum hitting with X (i.e. have the most members which intersect X). This answers a question of Barber, who extended previous results by Borg to characterise those sets X for which maximum hitting is achieved by the star.

Keywords

Set systems Intersecting families Compressions 

Notes

Acknowledgements

We thank Ben Barber for helpful discussions, and two anonymous reviewers for their helpful suggestions for improving the presentation of this manuscript.

Funding

Candida Bowtell was funded by London Mathematical Society (No. URB 14-30).

References

  1. 1.
    Ahlswede, R., Khachatrian, L.: The complete intersection theorem for systems of finite sets. Eur. J. Combin. 18, 125–136 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barber, B.: Maximum hitting for \(n\) sufficiently large. Graphs Combin. 30, 267–274 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borg, P.: Maximum hitting of a set by compressed intersecting families. Graphs Combin. 27, 785–797 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Q. J. Math. Oxf. 12, 313–320 (1961)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Frankl, P.: The Shifting Technique in Extremal Set Theory, Surveys in Combinatorics, pp. 81–110. Cambridge University Press, Cambridge (1987)Google Scholar
  6. 6.
    Han, J., Kohayakawa, Y.: The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton–Milner family. Proc. Am. Math. Soc. 145, 73–87 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hilton, A., Milner, E.: Some intersection theorems for systems of finite sets. Q. J. Math. Oxf. 18, 369–384 (1967)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.School of MathematicsUniversity of BirminghamBirminghamUK

Personalised recommendations