Classification of Maximum Hittings by Large Families

  • Candida Bowtell
  • Richard MycroftEmail author
Original Paper


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.


Set systems Intersecting families Compressions 



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


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


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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

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