Combinatorica

, Volume 3, Issue 3, pp 381–392

Extremal problems in discrete geometry

  • E. Szemerédi
  • W. T. TrotterJr.
Article

DOI: 10.1007/BF02579194

Cite this article as:
Szemerédi, E. & Trotter, W.T. Combinatorica (1983) 3: 381. doi:10.1007/BF02579194

Abstract

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 constantc1 so that when\(\sqrt n \leqq t \leqq \left( {_2^n } \right)\), the number of incidences betweenn points andt lines is less thanc1n2/3t2/3. Using this result, it follows immediately that there exists an absolute constantc2 so that ifk≦√n, then the number of lines containing at leastk points is less thanc2n2/k3. We then prove that there exists an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points. Finally, we show that there is an absolute constantc4 so that there are less than exp (c4n) sequences 2≦y1y2≦...≦yr for which there is a set ofn points and a setl1,l2, ...,lt oft lines so thatlj containsyj points.

AMS subject classification (1980)

51 M 0505 C 35

Copyright information

© Akadémiai Kiadó 1983

Authors and Affiliations

  • E. Szemerédi
    • 1
  • W. T. TrotterJr.
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary
  2. 2.Dept. of Mathematics and StatisticsUniversity of South CarolinaColumbiaUSA