A Comment on Ryser's Conjecture for Intersecting Hypergraphs

Let τ(H) be the cover number and ν(H) be the matching number of a hypergraph H. Ryser conjectured that every r-partite hypergraph H satisfies the inequality τ(H) ≤ (r − 1)ν(H). This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with ν(H) = 1, Ryser’s conjecture reduces to τ(H) ≤ 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(H) = 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(H) ≥ 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.