Largest size and union of Helly families

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A finite set system (hypergraph)H is said to have the Helly property if the members of each intersecting subsystem H′ of H share an element (i.e. H∩H′≠øFS for all H,H′∈H′ implies ∩H∈H′ H≠ø); H is k-uniform if |H|=k for all H∈H. In a previous paper, we con jectured that the union of t k-uniform Helly families (k≥3) on the same n-element set can have at most ∑t i=1(n-i k-1) members. Here we prove this conjecture for every k and every t ≤2k-2, for n sufficiently large with respect to k. The main tool is a result (an analogue of the Hilton-Milner theorem on intersecting k-uniform set systems) stating that if the sets in a k-uniform Helly family on n points have an empty intersection, then for large n, |H|≤(n-k-1 k-1)+(n-2 k-2)+1 , and the set system attaining equality is uniq ue. We also show that ∑t i=1(n-i k-1) is a (best possible) upper bound for the union of t intersecting k-uniform set systems on the same n-element set, whenever n≥(t+2k-1)(k-1). This result strengthens a particular case of the Hajnal-Rothschild theorem.

Original languageEnglish
Pages (from-to)319-327
Number of pages9
JournalDiscrete Mathematics
Issue number1-3
Publication statusPublished - Mar 15 1994

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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