Graphs and Combinatorics

, Volume 30, Issue 2, pp 267–274 | Cite as

Maximum Hitting for n Sufficiently Large

Original Paper
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Abstract

For a left-compressed intersecting family \({\fancyscript{A} \subseteq[n]^{(r)}}\) and a set \({X \subseteq [n]}\) , let \({\fancyscript{A}(X) = \{A \in \fancyscript{A} : A \cap X \neq \emptyset\}}\) . Borg asked: for which X is \({|\fancyscript{A}(X)|}\) maximised by taking \({\fancyscript{A}}\) to be all r-sets containing the element 1? We determine exactly which X have this property, for n sufficiently large depending on r.

Keywords

Intersecting family Compression Generating set Erdős–Ko–Rado theorem 

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References

  1. 1.
    Ahlswede, R., Khachatrian, L.H.: The complete intersection theorem for systems of finite sets. Eur. J. Combin. 18(2), 125–136 (1997). doi:10.1006/eujc.1995.0092 Google Scholar
  2. 2.
    Borg P.: Maximum hitting of a set by compressed intersecting families. Graphs Combin. 27(6), 785–797 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2) 12, 313–320 (1961)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Frankl, P.: The shifting technique in extremal set theory. In: Whitehead, C. (ed.) Surveys in Combinatorics, London Mathematical Society. Lecture Notes Series, vol. 123, pp. 81–110. Cambridge University Press, Cambridge (1987)Google Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsCentre for Mathematical SciencesCambridgeUK

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