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

Article

- Received:

DOI: 10.1007/BF02579184

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

## Abstract

Let*S* be a set of*n* non-collinear points in the Euclidean plane. It will be shown here that for some point of*S* the number of*connecting lines* through it exceeds*c · 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 most*n−x* points of*S*, then the number of connecting lines determined by*S* is greater than*c · x · n*.

### AMS subject classification (1980)

51 M 0505 C 35## Copyright information

© Akadémiai Kiadó 1983