Distinct distances in finite planar sets

P. Erdős, Peter Fishburn

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)97-132
Number of pages36
JournalDiscrete Mathematics
Volume175
Issue number1-3
Publication statusPublished - Oct 15 1997

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Distinct
F-vector
Minimise
Configuration
Subset
Triangular Lattice
Denote
Integer

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Erdős, P., & Fishburn, P. (1997). Distinct distances in finite planar sets. Discrete Mathematics, 175(1-3), 97-132.

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

In: Discrete Mathematics, Vol. 175, No. 1-3, 15.10.1997, p. 97-132.

Research output: Contribution to journalArticle

Erdős, P & Fishburn, P 1997, 'Distinct distances in finite planar sets', Discrete Mathematics, vol. 175, no. 1-3, pp. 97-132.
Erdős P, Fishburn P. Distinct distances in finite planar sets. Discrete Mathematics. 1997 Oct 15;175(1-3):97-132.
Erdős, P. ; Fishburn, Peter. / Distinct distances in finite planar sets. In: Discrete Mathematics. 1997 ; Vol. 175, No. 1-3. pp. 97-132.
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