Extremal problems in discrete geometry


In this paper, we establish several theorems involving configurations of points and lines in the Euclidean plane. Our results answer questions and settle conjectures of P. Erdõs, G. Purdy, and G. Dirac. The principal result is that there exists an absolute constantc 1 so that when\(\sqrt n \leqq t \leqq \left( {_2^n } \right)\), the number of incidences betweenn points andt lines is less thanc 1 n 2/3 t 2/3. Using this result, it follows immediately that there exists an absolute constantc 2 so that ifk≦√n, then the number of lines containing at leastk points is less thanc 2 n 2/k 3. We then prove that there exists an absolute constantc 3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc 3 n of the lines determined by then points. Finally, we show that there is an absolute constantc 4 so that there are less than exp (c 4n) sequences 2≦y 1y 2≦...≦y r for which there is a set ofn points and a setl 1,l 2, ...,l t oft lines so thatl j containsy j points.

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


  1. [1]

    J. Beck, On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry,Combinatorica 3 (3–4) (1983), 281–297.

    MATH  MathSciNet  Google Scholar 

  2. [2]

    P. Erdős, On some problems of elementary and combinatorial geometry,Annali di Mat., Ser. 4,103 (1974), 99–108.

    Article  Google Scholar 

  3. [3]

    P. Erdős, Some combinatorial problems in geometry,Conference held in Haifa, Israel, 1979, Lecture Notes in Mathematics,792, Springer, 1980, 46–53.

  4. [4]

    P. Erdős, Some applications of graph theory and combinatorial methods to number theory and geometry.Algebraic methods in graph theory, Coll. Math. Soc. T. Bolyai 25 (1981), 131–148.

    Google Scholar 

  5. [5]

    P. Erdős, Problems and results in combinatorial geometry,Proceedings of New York Academy of Science, to appear.

  6. [6]

    William Moser,Research Problems in Discrete Geometry, Mimeograph Notes (1981).

  7. [7]

    E. Szemerédi andWilliam T. Trotter, Jr., A Combinatorial Distinction Between the Euclidean and Projective Planes,submitted.

Download references

Author information



Additional information

Dedicated to Paul Erdős on his seventieth birthday

Research supported in part by NSF Grants ISP-8011451 and MCS-8202172.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Szemerédi, E., Trotter, W.T. Extremal problems in discrete geometry. Combinatorica 3, 381–392 (1983). https://doi.org/10.1007/BF02579194

Download citation

AMS subject classification (1980)

  • 51 M 05
  • 05 C 35