Almost Sharp Bounds on the Number of Discrete Chains in the Plane

Authors Nóra Frankl, Andrey Kupavskii

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Author Details

Nóra Frankl
  • Department of Mathematics, London School of Economics and Political Science, UK
  • Laboratory of Combinatorial and Geometric Structures at MIPT, Moscow, Russia
Andrey Kupavskii
  • Moscow Institute of Physics and Technology, Moscow, Russia
  • Institute for Advanced Study, Princeton, NJ, US
  • G-SCOP, CNRS, Grenoble, France


We thank Konrad Swanepoel and the referees for helpful comments on the manuscript.

Cite AsGet BibTex

Nóra Frankl and Andrey Kupavskii. Almost Sharp Bounds on the Number of Discrete Chains in the Plane. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ=(δ₁,… ,δ_k) of k distances, a (k+1)-tuple (p₁,… ,p_{k+1}) of distinct points in ℝ^d is called a (k,δ)-chain if ‖p_j-p_{j+1}‖ = δ_j for every 1 ≤ j ≤ k. What is the maximum number C_k^d(n) of (k,δ)-chains in a set of n points in ℝ^d, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • unit distance problem
  • unit distance graphs
  • discrete chains


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