Combinatorica

, Volume 3, Issue 3, pp 281–297

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

Authors

  • József Beck
    • Mathematical Institute of the Hungarian Academy of Sciences
Article

DOI: 10.1007/BF02579184

Cite this article as:
Beck, J. Combinatorica (1983) 3: 281. doi:10.1007/BF02579184

Abstract

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 0505 C 35

Copyright information

© Akadémiai Kiadó 1983