Triangles in arrangements of lines. II

Abstract

We show that given n lines in the real projective plane, no n − 1 n - 1 of which are concurrent, the number p 3 {p_3} of triangular regions formed is at most 2 5 n ( n − 1 ) \tfrac {2}{5}n(n - 1) , equality being possible. We also show that if n ⩾ 6 n \geqslant 6 then p 3 ⩽ 7 18 n ( n − 1 ) + 1 3 {p_3} \leqslant \tfrac {7}{{18}}n(n - 1) + \tfrac {1}{3} . Grünbaum has conjectured p 3 ⩽ 1 3 n ( n − 1 ) {p_3} \leqslant \tfrac {1}{3}n(n - 1) .