Israel Journal of Mathematics

, Volume 51, Issue 1–2, pp 79–89 | Cite as

Families of finite sets in which no set is covered by the union ofr others

  • P. Erdös
  • P. Frankl
  • Z. Füredi
Article

Abstract

Letfr(n, k) denote the maximum number ofk-subsets of ann-set satisfying the condition in the title. It is proved that
$$f_1 (n,r(t - 1) + 1 + d)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } (_{ t}^{n - d} )/(_{ t}^{k - d} )$$
wheneverd=0, 1 ordr/2t2 with equality holding iff there exists a Steiner systemS(t, r(t−1)+1,n−d). The determination offr(n, 2r) led us to a new generalization of BIBD (Definition 2.4). Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family.

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Copyright information

© Hebrew University 1985

Authors and Affiliations

  • P. Erdös
    • 1
  • P. Frankl
    • 2
  • Z. Füredi
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of ScienceHungary
  2. 2.CNRSParisFrance

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