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.