A linear system is a pair (P, L) where L is a family of subsets on a ground finite set P , such that |l ∩ l | ≤ 1, for every l, l ∈ L. The elements of P and L are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point. A subset T of points of P is a transversal of (P, L) if T intersects any line, and the transversal number, τ (P, L), is the minimum order of a transversal. On the other hand, a 2-packing set of a linear system (P, L) is a set R of lines, such that any three of them have a common point, then the 2packing number of (P, L), ν2(P, L), is the size of a maximum 2-packing set. It is known that the transversal number τ (P, L) is bounded above by a quadratic function of ν2 (P, L). An open problem is to haracterize the families of linear systems which satisfies τ (P, L) ≤ λν2(P, L), for some λ ≥ 1. In this paper, we give an infinite family of linear systems (P, L) which satisfies τ (P, L) = ν2(P, L) with smallest possible cardinality of L, as well as some properties of r-uniform intersecting linear systems (P, L), such that τ (P, L) = ν2(P, L) = r. Moreover, we state a characterization of 4-uniform intersecting linear systems (P, L) with τ (P, L) = ν2(P, L) = 4.
