Crossing edges and faces of line arrangements in the plane

Abstract

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