### Abstract

Let n and m be integers with n = m^{2} + m + 1. Then the projective plane of order m has n points and n lines with each line containing m + ≈ n^{1/2} points. In this paper, we consider the analogous problem for the Euclidean plane and show that there cannot be a comparably large collection of lines each of which contains approximately n^{1/2} points from a given set of n points. More precisely, we show that for every δ 〉 0, there exist constants c, n_{0} so that if n ⩾ n_{0}, it is not possible to find n points in the Euclidean plane and a collection of at least cn^{1/2} lines each containing at least δn^{1/2} of the points. This theorem answers a question posed by P. Erdös. The proof involves a covering lemma, which may be of independent interest, and an application of the first author's regularity lemma.

Original language | English |
---|---|

Pages (from-to) | 385-394 |

Number of pages | 10 |

Journal | European Journal of Combinatorics |

Volume | 4 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 1 1983 |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*European Journal of Combinatorics*,

*4*(4), 385-394. https://doi.org/10.1016/S0195-6698(83)80036-5

**A Combinatorial Distinction Between the Euclidean and Projective Planes.** / Szemerédi, E.; Trotter, W. T.

Research output: Contribution to journal › Article

*European Journal of Combinatorics*, vol. 4, no. 4, pp. 385-394. https://doi.org/10.1016/S0195-6698(83)80036-5

}

TY - JOUR

T1 - A Combinatorial Distinction Between the Euclidean and Projective Planes

AU - Szemerédi, E.

AU - Trotter, W. T.

PY - 1983/1/1

Y1 - 1983/1/1

N2 - Let n and m be integers with n = m2 + m + 1. Then the projective plane of order m has n points and n lines with each line containing m + ≈ n1/2 points. In this paper, we consider the analogous problem for the Euclidean plane and show that there cannot be a comparably large collection of lines each of which contains approximately n1/2 points from a given set of n points. More precisely, we show that for every δ 〉 0, there exist constants c, n0 so that if n ⩾ n0, it is not possible to find n points in the Euclidean plane and a collection of at least cn1/2 lines each containing at least δn1/2 of the points. This theorem answers a question posed by P. Erdös. The proof involves a covering lemma, which may be of independent interest, and an application of the first author's regularity lemma.

AB - Let n and m be integers with n = m2 + m + 1. Then the projective plane of order m has n points and n lines with each line containing m + ≈ n1/2 points. In this paper, we consider the analogous problem for the Euclidean plane and show that there cannot be a comparably large collection of lines each of which contains approximately n1/2 points from a given set of n points. More precisely, we show that for every δ 〉 0, there exist constants c, n0 so that if n ⩾ n0, it is not possible to find n points in the Euclidean plane and a collection of at least cn1/2 lines each containing at least δn1/2 of the points. This theorem answers a question posed by P. Erdös. The proof involves a covering lemma, which may be of independent interest, and an application of the first author's regularity lemma.

UR - http://www.scopus.com/inward/record.url?scp=85008387823&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85008387823&partnerID=8YFLogxK

U2 - 10.1016/S0195-6698(83)80036-5

DO - 10.1016/S0195-6698(83)80036-5

M3 - Article

AN - SCOPUS:85008387823

VL - 4

SP - 385

EP - 394

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 4

ER -