A theorem on arrangements of lines in the plane

AbstractLetA be an arrangement ofn lines in the plane. IfR1, …,Rr arer distinct regions ofA, andRi is api-gon (i=1, …,r) then we show that $$\sum\limits_{i = 1}^r {P_i \leqq n + 4} \left( {_2^r } \right)$$ . Further we show that for allr this bound is the best possible ifn is sufficiently large.