  A famous conjecture of Ryser is that in an $r$-partite hypergraph the
covering number is at most $r-1$ times the matching number. If true, this is
known to be sharp for $r$ for which there exists a projective plane of order
$r-1$. We show that the conjecture, if true, is also sharp for the smallest
previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal
number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has
covering number at least $r-1$. We find that $f(r)$ is achieved only by linear
hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}$. We
also improve the general lower bound on $f(r)$, showing that $f(r)\ge
3.052r+O(1)$.
  We show that a stronger form of Ryser's conjecture that was used to prove the
$r=3$ case fails for all $r>3$. We also prove a fractional version of the
following stronger form of Ryser's conjecture: in an $r$-partite hypergraph
there exists a set $S$ of size at most $r-1$, contained either in one side of
the hypergraph or in an edge, whose removal reduces the matching number by 1.
