On transversal and $2$-packing numbers in straight line systems on $\mathbb{R}^{2}$

A linear system is a pair $(X,\mathcal{F})$ where $\mathcal{F}$ is a finite family of subsets on a ground set $X$, and it satisfies that $|A\cap B|\leq 1$ for every pair of distinct subsets $A,B \in \mathcal{F}$. As an example of a linear system are the straight line systems, which family of subsets are straight line segments on $\mathbb{R}^{2}$. By $\tau$ and $\nu_2$ we denote the size of the minimal transversal and the 2--packing numbers of a linear system respectively. A natural problem is asking about the relationship of these two parameters; it is not difficult to prove that there exists a quadratic function $f$ holding $\tau\leq f(\nu_2)$. However, for straight line system we believe that $\tau\leq\nu_2-1$. In this paper we prove that for any linear system with $2$-packing numbers $\nu_2$ equal to $2, 3$ and $4$, we have that $\tau\leq\nu_2$. Furthermore, we prove that the linear systems that attains the equality have transversal and $2$-packing numbers equal to $4$, and they are a special family of linear subsystems of the projective plane of order $3$. Using this result we confirm that all straight line systems with $\nu_2\in\{2,3,4\}$ satisfies $\tau\leq\nu_2-1$.