Triangles in arrangements of points and lines in the plane

A conjecture of de Caen and Székely [D. de Caen, L.A. Székely, J. Combin. Theory Ser. A 77 (1997) 268–278] asserts that every arrangement of n points and m lines in the plane contains at most O ( n m ) triangles with vertices in the points and sides on the lines of the arrangement. We disprove the conjecture by constructing an arrangement of n points and n lines with Ω ( n 2 log log n ) such triangles.