A comment on Ryser's conjecture for intersecting hypergraphs

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Let τ(script H sign) be the cover number and ν({script H sign) be the matching number of a hypergraph script H sign. Ryser conjectured that every r-partite hypergraph script H sign satisfies the inequality τ(script H sign) ≤ (r-1) ν ({script H sign). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with ν(script H sign) = 1, Ryser's conjecture reduces to τ(script H sign) ≤ r-1. Even this conjecture is extremely difficult and is open for all r ≥ 6. For infinitely many r there are examples of intersecting r-partite hypergraphs with τ(script H sign) = r-1, demonstrating the tightness of the conjecture for such r. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting r-partite hypergraph be, given that its cover number is as large as possible, namely τ(script H sign) ≥ r-1? In this paper we solve this question for r ≤ 5, give an almost optimal construction for r = 6, prove that any r-partite intersecting hypergraph with τ(H) ≥ r - 1 must have at least (3- 1/√18)r(1-o(1)) ≈ 2.764r(1-o(1)) edges, and conjecture that there exist constructions with Θ(r) edges.

Original languageEnglish
Pages (from-to)101-109
Number of pages9
JournalGraphs and Combinatorics
Issue number1
StatePublished - May 2009


  • Covering
  • Hypergraph
  • Intersecting
  • Matching
  • R -partite
  • Ryser's conjecture

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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