### 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 constant c_{ 1} so that when {Mathematical expression}, the number of incidences between n points and t lines is less than c_{ 1} n^{ 2/3} t^{ 2/3}. Using this result, it follows immediately that there exists an absolute constant c_{ 2} so that if k≦√n, then the number of lines containing at least k points is less than c_{ 2} n^{ 2}/k^{ 3}. We then prove that there exists an absolute constant c_{ 3} so that whenever n points are placed in the plane not all on the same line, then there is one point on more than c_{ 3} n of the lines determined by the n points. Finally, we show that there is an absolute constant c_{ 4} so that there are less than exp (c_{ 4} √n) sequences 2≦y_{ 1}≦y_{ 2}≦...≦y_{ r} for which there is a set of n points and a set l_{ 1}, l_{ 2}, ..., l_{ t} of t lines so that l_{ j} contains y_{ j} points.

Original language | English |
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Pages (from-to) | 381-392 |

Number of pages | 12 |

Journal | Combinatorica |

Volume | 3 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - szept. 1 1983 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*3*(3-4), 381-392. https://doi.org/10.1007/BF02579194