Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2010), pp. 32-36
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I. N. Shnurnikov. Into how many regions do $n$ lines divide the plane if at most $n-k$ of them are concurrent?. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2010), pp. 32-36. http://geodesic.mathdoc.fr/item/VMUMM_2010_5_a5/
@article{VMUMM_2010_5_a5,
author = {I. N. Shnurnikov},
title = {Into how many regions do $n$ lines divide the plane if at most $n-k$ of them are concurrent?},
journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
pages = {32--36},
year = {2010},
number = {5},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VMUMM_2010_5_a5/}
}
TY - JOUR
AU - I. N. Shnurnikov
TI - Into how many regions do $n$ lines divide the plane if at most $n-k$ of them are concurrent?
JO - Vestnik Moskovskogo universiteta. Matematika, mehanika
PY - 2010
SP - 32
EP - 36
IS - 5
UR - http://geodesic.mathdoc.fr/item/VMUMM_2010_5_a5/
LA - ru
ID - VMUMM_2010_5_a5
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%0 Journal Article
%A I. N. Shnurnikov
%T Into how many regions do $n$ lines divide the plane if at most $n-k$ of them are concurrent?
%J Vestnik Moskovskogo universiteta. Matematika, mehanika
%D 2010
%P 32-36
%N 5
%U http://geodesic.mathdoc.fr/item/VMUMM_2010_5_a5/
%G ru
%F VMUMM_2010_5_a5
A number of connective components of the real projective plane, disjoint with the family of $n\geq 2$ distinct lines is estimated provided at most $n-k$ lines are concurrent. If $n\geq\frac{k^2+k}2+3$, then the number of regions is at least $(k+1)(n-k)$. Thus, a new proof of 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.
