The transversal numbers of linear systems with given 2-packing numbers

A linear system is a pair ( X , F ) where F is a finite family of subsets on a ground set X and it satisfies that | A ∩ B | ⩽ 1 for every pair of distinct subsets A , B ∈ F . We study the relation between two parameters in linear systems: the transversal and the 2-packing numbers. Our main theorem shows that any linear system with 2-packing number equal to k has transversal number at most k, for k = { 2 , 3 , 4 } , and for k = 4 the equality is attained only for one special family of linear subsystems of the projective plane of order 4.