, Volume 3, Issue 3–4, pp 281–297 | Cite as

On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry

  • József Beck


LetS be a set ofn non-collinear points in the Euclidean plane. It will be shown here that for some point ofS the number ofconnecting lines through it exceedsc · n. This gives a partial solution to an old problem of Dirac and Motzkin. We also prove the following conjecture of Erdős: If any straight line contains at mostn−x points ofS, then the number of connecting lines determined byS is greater thanc · x · n.

AMS subject classification (1980)

51 M 05 05 C 35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Beck andJ. Spencer, Unit distances, submitted toJournal of Combinatorial Theory, Series A (1982)Google Scholar
  2. [2]
    H. S. M. Coxeter,Introduction to geometry, John Wiley and Sons, New York, 1961.zbMATHGoogle Scholar
  3. [3]
    G. A. Dirac, Collinearity properties of sets of points,Quart. J. Math. 2 (1951) 221–227.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    P. Erdős, On some problems of elementary and combiratorial geometry,Annali di Mat. Pura et Applicata, Ser. IV. 103 (1975) 99–108.CrossRefGoogle Scholar
  5. [5]
    P. Erdős, Some applications of graph theory and combinatorial methods to number theory and geometry,Colloquia Math. Soc. János Bolyai, Algebraic methods in graph theory, Szeged (Hungary) (1978) 137–148.Google Scholar
  6. [6]
    P. Erdős, On the combinatorial problems which I would most like to see solved,Combinatorica 1 (1981) 25–42.MathSciNetGoogle Scholar
  7. [7]
    B. Grünbaum,Arrangements and spreads, Regional Conference Series in Mathematics 10, Amer. Math. Soc., 1972.Google Scholar
  8. [8]
    E. Jucovič, Problem 24,Combinatorial structures and their applications, Gordon and Breach, New York, 1970.Google Scholar
  9. [9]
    L. M. Kelly andW. Moser, On the number of ordinary lines determined byn points,Canad. J. Math. 10 (1958) 210–219.zbMATHMathSciNetGoogle Scholar
  10. [10]
    W. Moser,Research problems in discrete geometry, Mimeograph notes, 1981.Google Scholar
  11. [11]
    T. S. Motzkin, The lines and planes connecting the points of a finite set,Trans. Amer. Math. Soc. 70 (1951) 451–464.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    E. Szemerédi andW. T. Trotter, Extremal problems in discrete geometry,Combinatorica 3 (3–4) (1983) 381–392.zbMATHMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1983

Authors and Affiliations

  • József Beck
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations