Crossing edges and faces of line arrangements in the plane

Abstract

For any natural number $$n$$n we define $$f(n)$$f(n) to be the minimum number with the following property. Given any arrangement $$\mathcal{A}(\mathcal{L})$$A(L) of $$n$$n blue lines in the real projective plane one can find $$f(n)$$f(n) red lines different from the blue lines such that any edge in the arrangement $$\mathcal{A}(\mathcal{L})$$A(L) is crossed by a red line. We define $$h(n)$$h(n) to be the minimum number with the following property. Given any arrangement $$\mathcal{A}(\mathcal{L})$$A(L) of $$n$$n blue lines in the real projective plane one can find $$h(n)$$h(n) red lines different from the blue lines such that every face in the arrangement $$\mathcal{A}(\mathcal{L})$$A(L) is crossed in its interior by a red line. In this paper we show $$f(n)=2n-o(n)$$f(n)=2n-o(n) and $$h(n)=n-o(n)$$h(n)=n-o(n).