A Comment on Ryser’s Conjecture for Intersecting Hypergraphs

AbstractLet $$\tau({\mathcal{H}})$$ be the cover number and $$\nu({\mathcal{H}})$$ be the matching number of a hypergraph $${\mathcal{H}}$$. Ryser conjectured that every r-partite hypergraph $${\mathcal{H}}$$ satisfies the inequality $$\tau({\mathcal{H}}) \leq (r-1) \nu ({\mathcal{H}})$$. This conjecture is open for all r ≥ 4. For intersecting hypergraphs, namely those with $$\nu({\mathcal{H}}) = 1$$, Ryser’s conjecture reduces to $$\tau({\mathcal{H}}) \leq r-1$$. 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 $$\tau({\mathcal{H}}) = r-1$$, 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 $$\tau({\mathcal{H}}) \ge r-1$$? 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 $$(3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))$$ edges, and conjecture that there exist constructions with Θ(r) edges.