, Volume 7, Issue 4, pp 393-397

A theorem on arrangements of lines in the plane

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Abstract

LetA be an arrangement ofn lines in the plane. IfR 1, …,R r arer distinct regions ofA, andR i is ap i-gon (i=1, …,r) then we show that \(\sum\limits_{i = 1}^r {P_i \leqq n + 4} \left( {_2^r } \right)\) . Further we show that for allr this bound is the best possible ifn is sufficiently large.

Financial support for this research was provided by the Carnegie Trust for the Universities of Scotland.