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
Voir la notice de l'article provenant de la source Math-Net.Ru
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.
@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},
publisher = {mathdoc},
number = {5},
year = {2010},
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 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/VMUMM_2010_5_a5/ LA - ru ID - VMUMM_2010_5_a5 ER -
%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 %I mathdoc %U http://geodesic.mathdoc.fr/item/VMUMM_2010_5_a5/ %G ru %F VMUMM_2010_5_a5
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/