The Maximum Number of Triangles in Arrangements of Pseudolines

Grünbaum has conjectured that any arrangement ofnpseudolines in the real projective plane has at most 13n(n−1) triangular faces whennis sufficiently large. We prove this conjecture forn⩾9; the result does not hold forn⩽8. The structure of extremal examples is explored and an infinite family of non simple arrangements with 13n(n−1) triangles is constructed. As an application, we show that the number of simplices in arrangements ofn⩾10 pseudoplanes is always less than[formula].