Crossing by lines all edges of a line arrangement

Let L be a family of n blue lines in the real projective plane. Suppose that R is a collection of m red lines, different from the blue lines, and that every edge in the arrangement A ( L ) is crossed by a line in R . We show that m ≥ n − 1 3.5 . Our result is more general, and applies to pseudo-line arrangements A ( L ) , and even weaker assumptions are required for R . 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 n nonconcurrent lines in the plane. We draw a possible relation between the two problems.