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


Letf r(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/2t 2 with equality holding iff there exists a Steiner systemS(t, r(t−1)+1,n−d). The determination off r(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.


Pairwise Disjoint Maximum Cardinality Intersection Theorem Steiner System Element Subset 
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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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