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 segments of $$\mathbb {R}^2$$ and where any two lines share a point. Recently, the author [A. Vazquez-Avila, On intersecting straight line systems, J. Discret. Math. Sci. Cryptogr. Accepted] proved that any intersecting r-uniform straight line system $$(P,\mathcal {L})$$ with $$r\ge \nu _2$$ has transversal number at most $$\nu _2-1$$ , 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. This paper improves such upper bound if the intersecting r-uniform straight line system satisfies $$r=\nu _2$$ . Also, those results have immediate consequences for some questions given by Oliveros et al. in [D. Oliveros, C. O’Neill and S. Zerbib, The geometry and combinatorics of discrete line segment hypergraphs, Discrete Math. 343 (2020), no. 6, 111825].