The maximum number of triangles in arrangements of pseudolines (Q1907111)

The main result of this paper is the proof of the following conjecture of Grünbaum: any arrangement of \(n\geq 9\) pseudolines in the real projective plane has at most \({1 \over 3} n(n - 1)\) triangular faces. The result does not hold for \(n \leq 8\). The structure of extremal examples is explored and an infinite family of non simple arrangements with \({1 \over 3} n(n - 1)\) triangles is constructed. As an application, it is shown that the number of simplices in arrangements of \(n \geq 10\) pseudoplanes is always less than \({1 \over 12} n(n - 1) (n - 2)\).