Journal of Geometry

, Volume 64, Issue 1, pp 89-94

First online:

Triangles in arrangements of lines

  • Haiwei GuAffiliated withDepartment of Mathematics, Qufu Normal University

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Givenn lines in the real projective plane, Grünbaum conjectures that, for n≥16, the numberp 3 of triangular regions determined by the lines is at most 1/3n(n−1). We show that ifn≥7 thenp 3 ≤8/21n(n−1)+2/7, we also point out that if no vertex is a point of intersection of exactly three of the lines, thenp 3≤1/3n(n−1).