A note on extremal intersecting linear Ryser systems

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.