A comment on Ryser's conjecture for intersecting hypergraphs

Let $\tau (\mathcal{H})$ 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 $\tau (\mathcal{H})\le (r-1)\nu (\mathcal{H})$. This conjecture is open for all $r\ge 4$. For intersecting hypergraphs, namely those with $\nu (\mathcal{H})=1$, Ryser’s conjecture reduces to $\tau($\mathcal{H}$) \leq r-1$\mathrm{.}Eventhisconjectureisextremelydifficultandisopenforall$ r \ge 6$\mathrm{.}Forinfinitelymany$r$thereareexamplesofintersecting$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 $r \le 5$,giveanalmostoptimalconstructionfor$r=6$,provethatany$r$-partiteintersectinghypergraphwith$\tau(H) \ge r-1$musthaveatleast$(3-\frac{1}{\sqrt{18}})r(1-o(1)) \approx 2.764r(1-o(1))$ edges, and conjecture that there exist constructions with $\mathrm{\Theta }(r)$ edges.