  A famous conjecture of Ryser states that any $r$-partite set system has
transversal number at most $r-1$ times their matching number. This conjecture
is only known to be true for $r\leq3$ in general, for $r\leq5$ if the set
system is intersecting, and for $r\leq9$ if the intersecting set system is
linear. In this note, we deal with Ryser's Conjecture for intersecting
$r$-partite linear systems; that is, if $\tau$ is the transversal number for an
intersecting $r$-partite linear system, then Ryser's Conjecture states that
$\tau\leq r-1$. If this conjecture is true, this is known to be sharp for $r$
for which there exists a projective plane of order $r-1$. There has also been
considerable effort to find intersecting $r$-partite set systems whose
transversal number is $r-1$.
  In this note, the following is proved: if $r\geq4$ is an even integer, then
$f_l(r)\geq3(r-2)+1$, where $f_l(r)$ is the minimum number of lines of an
intersecting $r$-partite linear system whose transversal number is $r-1$. This
lower bound gives an exact value for $f_l(r)$, for some small values of $r$.
Also, we prove that any $r$-partite linear system satisfies $\tau\leq r-1$ if
$\nu_2\leq r$ for all $r\geq3$ odd integer and $\nu_2\leq r-1$ for all $r\geq4$
even integer, where $\nu_2$ is the maximum cardinality of a subset of lines
$R\subseteq\mathcal{L}$ such that every triplet of different elements of $R$
does not have a common point.
