Into how many regions do n lines divide the plane if at most n − k of them are concurrent?
Igor' Nikolaevich Shnurnikov · Moscow University Mathematics Bulletin · 2010
The number of connected components of the complement in the real projective plane to a family of n ≥2 different lines such that any point belongs to at most n − k of them is estimated. If $$ n \geqslant \frac{{k^2 + k}} {2} + 3 $$ , then the number of regions is at least (k+1)(n−k). Thus, a new proof of N. Martinov’s theorem is obtained. This theorem determines all pairs of integers (n, f) such that there is an arrangement of n lines dividing the projective plane into f regions.