On the number of regions and multiplicities of vertices in plane   arrangements
  For an arrangement of $n$ pseudolines in the real projective plane let us denote by $t_i$ the number of vertices incident to $i$ lines. We obtain a linear on $t_i$ inequality similar to the Hirzebruch one, but with an elementary proof. We present an algorithm for producing lower bounds of the number of regions basing on linear on $t_i$ inequalities like the above-mentioned. Lower bounds arise in connection with Martinov theorem on the set of all possible numbers of regions and we show how the new bounds may be applied in it. 
Igor Shnurnikov
math.CO math.GT