### 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 |
---|---|

Pages (from-to) | 381-392 |

Number of pages | 12 |

Journal | Combinatorica |

Volume | 3 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Sep 1983 |

### Fingerprint

### Keywords

- AMS subject classification (1980): 51M05, 05C35

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Combinatorica*,

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

**Extremal problems in discrete geometry.** / Szemerédi, E.; Trotter, W. T.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 3, no. 3-4, pp. 381-392. https://doi.org/10.1007/BF02579194

}

TY - JOUR

T1 - Extremal problems in discrete geometry

AU - Szemerédi, E.

AU - Trotter, W. T.

PY - 1983/9

Y1 - 1983/9

N2 - 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.

AB - 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.

KW - AMS subject classification (1980): 51M05, 05C35

UR - http://www.scopus.com/inward/record.url?scp=51249182705&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249182705&partnerID=8YFLogxK

U2 - 10.1007/BF02579194

DO - 10.1007/BF02579194

M3 - Article

AN - SCOPUS:51249182705

VL - 3

SP - 381

EP - 392

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 3-4

ER -