### 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 |
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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 |

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### 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