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2011 | 7 (14) | 65-70
Article title

On some extremal problem in discrete geometry

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EN
Abstracts
EN
Let p, q, r be any three lines in the plane passing through a common point and suppose that O, P, Q, R are any four collinear points such that P  p, Q  q, R  r, P and R are harmonic conjugates with respect to O and Q (that is, |OP|/|PQ|=|OR|/|QR|). For every k  2, we construct a set Xn of n = 4k points, which is distributed on the lines p, q, r, but each element of Xn  {O} is incident to at most n/2 lines spanned by Xn  {O}.
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References
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  • Coxeter H.S. (1961). Introduction to Geometry. John Wiley and Sons. New York. Dirac G.A. (1951). Collinearity properties of sets of points. Quarterly J. Math. Vol. 2. Pp. 221-227.
  • Grünbaum B. (1972). Arrangements and spreads. Regional Conference Series in Mathematics. Vol. 10. Amer. Math. Soc.
  • Grünbaum B. (2010). Dirac’s conjecture concerning high-incidence elements in aggregates. Geombinatorics. Vol. 20. Pp. 48-55.
  • Motzkin T.S. (1951). The lines and planes connecting the points of a finite set. Trans. Amer. Math. Soc. Vol. 70. Pp. 451-464.
  • Szemerédi E., Trotter W.T. (1983). Extremal problems in discrete geometry. Combinatorica. Vol. 3(3-4). Pp. 381-392.
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Publication order reference
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YADDA identifier
bwmeta1.element.desklight-32e1a331-c4cd-4408-9ec3-be5b0d1a1b3b
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