  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
$\tau(\mathcal{H}) \leq (r-1) \nu (\mathcal{H})$. This conjecture is open for
all $r \ge 4$. For intersecting hypergraphs, namely those with
$\nu(\mathcal{H})=1$, Ryser's conjecture reduces to $\tau(\mathcal{H}) \leq
r-1$. Even this conjecture is extremely difficult and is open for all $ r \ge
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 $r
\le 5$, give an almost optimal construction for $r=6$, prove that any
$r$-partite intersecting hypergraph with $\tau(H) \ge r-1$ 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 $\Theta(r)$ edges.
