## Crossing numbers and hard Erdős problems in discrete geometry (1997)

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Venue: | COMBINATORICS, PROBABILITY AND COMPUTING |

Citations: | 109 - 1 self |

### BibTeX

@MISC{Székely97crossingnumbers,

author = {László A. Székely},

title = {Crossing numbers and hard Erdős problems in discrete geometry},

year = {1997}

}

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### Abstract

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

### Citations

143 |
Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput
- Clarkson, Edelsbrunner, et al.
- 1990
(Show Context)
Citation Context ...some of those theorems were given simpler proofs and generalizations which used methods from the theory of VC dimension and extremal graph theory (see Clarkson, Edelsbrunner, Guibas, Sharir and Welzl =-=[6]-=-, Füredi and Pach [11] and Pach and Agarwal [18], Pach and Sharir [19]), the simpler proofs still lacked the simplicity and generality shown here. I believe that the notion of crossing number is a cen... |

131 | On sets of distances of n points
- Erdős
- 1946
(Show Context)
Citation Context ...e we in the second, as l<4n/(k − 1) <cn 2 /k 3 . Theorem 3, which was conjectured by Erdős and Purdy [9], is known to be tight for 2 ≤ k ≤ √ n for the points of the √ n × √ n grid [2]. In 1946, Erdős =-=[8]-=- conjectured that the number of unit distances among n points is at most n 1+o(1) and proved that this number is at most cn 3/2 . In 1973, Józsa and Szemerédi [13] improved this bound to o(n 3/2 ). In... |

73 |
Crossing-free subgraphs
- Ajtai, Chvátal, et al.
- 1982
(Show Context)
Citation Context ...le.” (H. Steinhaus) The main aim of this paper is to derive short proofs for a number of theorems in discrete geometry from Theorem 1. Theorem 1. (Leighton [16], Ajtai, Chvátal, Newborn and Szemerédi =-=[1]-=-) For any simple graph G with n vertices and e ≥ 4n edges, the crossing number of G on the plane is at least e 3 /(100n 2 ). For many graphs, Theorem 1 is tight within a constant multiplicative factor... |

66 |
Crossing number problems
- Erdös, Guy
- 1973
(Show Context)
Citation Context ...≥ 4n edges, the crossing number of G on the plane is at least e 3 /(100n 2 ). For many graphs, Theorem 1 is tight within a constant multiplicative factor. This result was conjectured by Erdős and Guy =-=[10, 12]-=-, and first proved by Leighton [16], who was unaware of the conjecture, and independently by Ajtai, Chvátal, Newborn and Szemerédi [1]. The theorem is still hardly known – some distinguished mathemati... |

37 |
On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry
- Beck
- 1983
(Show Context)
Citation Context ...e at home, and so are we in the second, as l<4n/(k − 1) <cn 2 /k 3 . Theorem 3, which was conjectured by Erdős and Purdy [9], is known to be tight for 2 ≤ k ≤ √ n for the points of the √ n × √ n grid =-=[2]-=-. In 1946, Erdős [8] conjectured that the number of unit distances among n points is at most n 1+o(1) and proved that this number is at most cn 3/2 . In 1973, Józsa and Szemerédi [13] improved this bo... |

21 | The number of different distances determined by n points in the plane
- Chung
- 1984
(Show Context)
Citation Context ...st cn/ √ log n distinct distances, and showed that the minimum is at least √ n. Over the years, this lower bound has been improved several times. In 1952, Moser [17] proved n 2/3 , then in 1984 Chung =-=[5]-=- improved n 2/3 to n 5/7 , and in an unpublished manuscript, Beck [3] proved n 58/81−ε . The best lower bound to date has been proved by Chung, Szemerédi and Trotter [7]. Theorem 9. (Chung, Szemerédi ... |

21 |
Crossing number of graphs
- Guy, Alavi, et al.
- 1972
(Show Context)
Citation Context ...≥ 4n edges, the crossing number of G on the plane is at least e 3 /(100n 2 ). For many graphs, Theorem 1 is tight within a constant multiplicative factor. This result was conjectured by Erdős and Guy =-=[10, 12]-=-, and first proved by Leighton [16], who was unaware of the conjecture, and independently by Ajtai, Chvátal, Newborn and Szemerédi [1]. The theorem is still hardly known – some distinguished mathemati... |

19 |
The number of different distances determined by a set of points in the Euclidean plane, Discrete Comput
- Chung, Szemerédi, et al.
- 1992
(Show Context)
Citation Context ...ved n 2/3 , then in 1984 Chung [5] improved n 2/3 to n 5/7 , and in an unpublished manuscript, Beck [3] proved n 58/81−ε . The best lower bound to date has been proved by Chung, Szemerédi and Trotter =-=[7]-=-. Theorem 9. (Chung, Szemerédi and Trotter [7]) At least n 4/5 /(log n) c distinct distances are determined by n points in the plane. They proved this for a large c, and claimed that a much smaller c,... |

12 |
Problems and results in combinatorial geometry
- Erdős
(Show Context)
Citation Context ... lines is at most l greater than the number of edges in G. Theorem 1 finishes the proof: either 4n ≥ #i − l or cr(G) ≥ c(#i − l) 3 /n 2 . For n = l the statement of Theorem 2 was conjectured by Erdős =-=[9]-=-. Although Theorem 3 below is known to be a simple corollary of Theorem 2, we prove it, since we use it in the proof of Theorem 4. Theorem 3. (Szemerédi and Trotter [24]) Let 2 ≤ k ≤ √ n.Fornpoints in... |

11 | Traces of finite sets: extremal problems and geometric applications. Extremal problems for finite sets (Visegrád
- Füredi, Pach
- 1991
(Show Context)
Citation Context ...s were given simpler proofs and generalizations which used methods from the theory of VC dimension and extremal graph theory (see Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [6], Füredi and Pach =-=[11]-=- and Pach and Agarwal [18], Pach and Sharir [19]), the simpler proofs still lacked the simplicity and generality shown here. I believe that the notion of crossing number is a central one for discrete ... |

5 |
Unit distances
- Beck, Spencer
(Show Context)
Citation Context ...least √ n. Over the years, this lower bound has been improved several times. In 1952, Moser [17] proved n 2/3 , then in 1984 Chung [5] improved n 2/3 to n 5/7 , and in an unpublished manuscript, Beck =-=[3]-=- proved n 58/81−ε . The best lower bound to date has been proved by Chung, Szemerédi and Trotter [7]. Theorem 9. (Chung, Szemerédi and Trotter [7]) At least n 4/5 /(log n) c distinct distances are det... |

4 | On the number of dierent distances determined by n points in the plane - Chung |

4 | The number of dierent distances determined by a set of points in the Euclidean plane, Discrete Comp. Geometry 7 - Chung, Szemeredi, et al. - 1992 |

1 | Traces of nite sets: extremal problems and geometric applications, in: Extremal Problems for Finite Sets, Visegrad - Furedi, Pach - 1991 |