Crossing by lines all edges of a line arrangement

Let LL be a family of nn blue lines in the real projective plane. Suppose that RR is a collection of mm red lines, different from the blue lines, and that every edge in the arrangement A(L)A(L) is crossed by a line in RR. We show that m≥n−13.5. Our result is more general, and applies to pseudo-line arrangements A(L)A(L), and even weaker assumptions are required for RR. Our result is motivated by the famous conjecture of Dirac about the existence of a line with many intersection points on it in any arrangement of nn nonconcurrent lines in the plane. We draw a possible relation between the two problems.