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Into How Many Regions Do n Lines Divide the Plane If at Most n-k of Them Are Concurrent? / Shnurnikov I.N. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2010. № 5. P. 32-36 [Moscow Univ. Math. Bulletin. Vol. 65, No 5, 2010. P. 208-212]. 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≥(k2 + 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.
Key words: arrangements of lines, polygonal decompositions of projective plane.
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