On transversal and 2-packing numbers in uniform linear systems

Abstract 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 2-packing number of ( P , L ) , ν 2 ( 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 ν 2 ( P , L ) . An open problem is to characterize the families of linear systems which satisfies τ ( P , L ) ≤ λ ν 2 ( P , L ) , for some λ ≥ 1 . In this paper, we give an infinite family of linear systems ( P , L ) which satisfies τ ( P , L ) = ν 2 ( 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 ) = ν 2 ( P , L ) = r . Moreover, we state a characterization of 4-uniform intersecting linear systems ( P , L ) with τ ( P , L ) = ν 2 ( P , L ) = 4 .