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

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The following generalisation of the Erdős Unit Distance problem was recently suggested by Palsson, Senger, and Sheffer. For a fixed sequence δ = (δ1, …, δk) of k distances, a (k + 1)-tuple (p1, …, pk+1) of distinct points in ℝd is called a k-chain if ∥pjpj+1∥ = δj for every 1 ≤ jk. What is the maximum number C dk (n) of k-chains in a set of n points in ℝd? 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, and propose further generalisations.

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We thank Konrad Swanepoel and Peter Allen for helpful comments on the manuscript. We also thank Dömötör Pálvölgyi for suggesting Proposition 3.2. Research of Andrey Kupavskii is supported by the grant RSF N 21-71-10092,

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Correspondence to Andrey Kupavskii.

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Frankl, N., Kupavskii, A. Almost Sharp Bounds on the Number of Discrete Chains in the Plane. Combinatorica 42 (Suppl 1), 1119–1143 (2022).

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