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Nearly Equal Distances in the Plane

  • Paul Erdős (a1), Endre Makai (a1) and János Pach (a2)

For any positive integer k and ε > 0, there exist nk,ε, ck, e > 0 with the following property. Given any system of n > nk,ε points in the plane with minimal distance at least 1 and any t1, t2…, tk ≥ 1, the number of those pairs of points whose distance is between ti and for some 1 ≤ ik, is at most (n2/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.

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[9]Turán, P. (1941) Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436452. (Hungarian, German summary.)
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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