On transversal numbers of intersecting straight line systems and intersecting segment systems

An intersecting $r$-uniform straight line system is an intersecting linear system whose lines consist of $r$ points on straight line segment of $\mathbb{R}^2$ and any two lines share a point. Recently, the author [A. V\'azquez-\'Avila, \emph{On intersecting straight line systems}, J. Discret. Math. Sci. Cryptogr. Accepted] proved that any intersecting $r$-uniform straight line system $(P,\mathcal{L})$ has transversal number at most $\nu_2-1$, with $r\geq\nu_2$, 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. In this paper, we improve such upper bound if the intersecting $r$-uniform straight line system satisfies $r=\nu_2$. This result has immediate consequences for some questions given by Oliveros et al. [D. Oliveros, C. O'Neill and S. Zerbib, \emph{The geometry and combinatorics of discrete line segment hypergraphs}, Discrete Math. {\bf 343} (2020), no. 6, 111825].