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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Triangles in arrangements of lines. II


Author: George Purdy
Journal: Proc. Amer. Math. Soc. 79 (1980), 77-81
MSC: Primary 05B35; Secondary 52A40
MathSciNet review: 560588
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Abstract: We show that given n lines in the real projective plane, no $ n - 1$ of which are concurrent, the number $ {p_3}$ of triangular regions formed is at most $ \tfrac{2}{5}n(n - 1)$, equality being possible.

We also show that if $ n \geqslant 6$ then $ {p_3} \leqslant \tfrac{7}{{18}}n(n - 1) + \tfrac{1}{3}$. Grünbaum has conjectured $ {p_3} \leqslant \tfrac{1}{3}n(n - 1)$.


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DOI: https://doi.org/10.1090/S0002-9939-1980-0560588-4
Article copyright: © Copyright 1980 American Mathematical Society