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


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    J. Beck andJ. Spencer, Unit distances, submitted toJournal of Combinatorial Theory, Series A (1982)

  2. [2]

    H. S. M. Coxeter,Introduction to geometry, John Wiley and Sons, New York, 1961.

    Google Scholar 

  3. [3]

    G. A. Dirac, Collinearity properties of sets of points,Quart. J. Math. 2 (1951) 221–227.

    MATH  Article  MathSciNet  Google 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.

    Article  Google 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.

    MathSciNet  Google Scholar 

  7. [7]

    B. Grünbaum,Arrangements and spreads, Regional Conference Series in Mathematics 10, Amer. Math. Soc., 1972.

  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.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    W. Moser,Research problems in discrete geometry, Mimeograph notes, 1981.

  11. [11]

    T. S. Motzkin, The lines and planes connecting the points of a finite set,Trans. Amer. Math. Soc. 70 (1951) 451–464.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    E. Szemerédi andW. T. Trotter, Extremal problems in discrete geometry,Combinatorica 3 (3–4) (1983) 381–392.

    MATH  MathSciNet  Google Scholar 

Download references

Author information



Additional information

Dedicated to Paul Erdős on his seventieth birthday

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Beck, J. On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry. Combinatorica 3, 281–297 (1983). https://doi.org/10.1007/BF02579184

Download citation

AMS subject classification (1980)

  • 51 M 05
  • 05 C 35