### Abstract

For each n ≥ 3 let F_{n} denote the set of all integer vectors f = (f_{1},f_{2}, ... , f_{n}) with 1 ≤ f_{1} ≤ f_{2} ≤ ⋯ ≤ f_{n} ≤ n - 1 for which there exists a set {x_{1}, x_{2}, ... , x_{n}} of n points in the plane such that each x_{i} has exactly f_{i} different distances to the other n - 1 points. Thus, F_{3} = {(1,1,1), (1,2,2), (2,2,2)}. We determine all n-point configurations for n ≤ 7 that minimize Σf_{i} over F_{n} and show that min {Σf_{i}: f ∈ F_{8}} = 24. We note that every small n except n ∈ {8, 9} has a subset of the triangular lattice among its sum-minimizing configurations, and conjecture that subsets of the lattice minimize Σf_{i} for all large n.

Original language | English |
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Pages (from-to) | 97-132 |

Number of pages | 36 |

Journal | Discrete Mathematics |

Volume | 175 |

Issue number | 1-3 |

Publication status | Published - Oct 15 1997 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*175*(1-3), 97-132.

**Distinct distances in finite planar sets.** / Erdős, P.; Fishburn, Peter.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 175, no. 1-3, pp. 97-132.

}

TY - JOUR

T1 - Distinct distances in finite planar sets

AU - Erdős, P.

AU - Fishburn, Peter

PY - 1997/10/15

Y1 - 1997/10/15

N2 - For each n ≥ 3 let Fn denote the set of all integer vectors f = (f1,f2, ... , fn) with 1 ≤ f1 ≤ f2 ≤ ⋯ ≤ fn ≤ n - 1 for which there exists a set {x1, x2, ... , xn} of n points in the plane such that each xi has exactly fi different distances to the other n - 1 points. Thus, F3 = {(1,1,1), (1,2,2), (2,2,2)}. We determine all n-point configurations for n ≤ 7 that minimize Σfi over Fn and show that min {Σfi: f ∈ F8} = 24. We note that every small n except n ∈ {8, 9} has a subset of the triangular lattice among its sum-minimizing configurations, and conjecture that subsets of the lattice minimize Σfi for all large n.

AB - For each n ≥ 3 let Fn denote the set of all integer vectors f = (f1,f2, ... , fn) with 1 ≤ f1 ≤ f2 ≤ ⋯ ≤ fn ≤ n - 1 for which there exists a set {x1, x2, ... , xn} of n points in the plane such that each xi has exactly fi different distances to the other n - 1 points. Thus, F3 = {(1,1,1), (1,2,2), (2,2,2)}. We determine all n-point configurations for n ≤ 7 that minimize Σfi over Fn and show that min {Σfi: f ∈ F8} = 24. We note that every small n except n ∈ {8, 9} has a subset of the triangular lattice among its sum-minimizing configurations, and conjecture that subsets of the lattice minimize Σfi for all large n.

UR - http://www.scopus.com/inward/record.url?scp=0040954949&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040954949&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0040954949

VL - 175

SP - 97

EP - 132

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -