Crossing by lines all edges of a line arrangement

Abstract

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−13.5. 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.