Journal of Geometry

, Volume 64, Issue 1–2, pp 89–94 | Cite as

Triangles in arrangements of lines

  • Haiwei Gu
Article
  • 36 Downloads

Abstract

Givenn lines in the real projective plane, Grünbaum conjectures that, for n≥16, the numberp3 of triangular regions determined by the lines is at most 1/3n(n−1). We show that ifn≥7 thenp3 ≤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, thenp3≤1/3n(n−1).

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References

  1. [1]
    B. Grünbaum,Arrangements and Spreads, Amer. Math. Soc., Providence, R.I., 1972.Google Scholar
  2. [2]
    F. Levi, Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade,Ber. Math. Phys. Kl. SÄchs. Akad. Wiss. Leipzig 21 (1926), 256–267.Google Scholar
  3. [3]
    L. M. Kelly andW. O. J. Moser, On the number of ordinary lines determined byn points,Canad. J. Math. 10 (1958), 210–219.Google Scholar
  4. [4]
    G. Purdy, Triangles in arrangements of lines,Discrete Math. 25 (1959), 157–163.Google Scholar
  5. [5]
    G. Purdy, Triangles in arrangements of lines. II,Proc. Amer. Math. Soc. 79 (1980), 77–81.Google Scholar
  6. [6]
    T. O. Strommer, Triangles in arrangements of lines,J. Combin. Theory Ser. A 23, (1977), 314–320.Google Scholar

Copyright information

© BirkhÄuser Verlag 1999

Authors and Affiliations

  • Haiwei Gu
    • 1
  1. 1.Department of MathematicsQufu Normal UniversityQufu, ShandongPeople's Republic of China

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