The following generalisation of the Erdős Unit Distance problem was recently suggested by Palsson, Senger, and Sheffer. For a fixed 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-chain if ∥p_j − p_j+1∥ = δ_j for every 1 ≤ j ≤ k. What is the maximum number C ^d_k (n) of k-chains in a set of n points in ℝ^d? 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.

Palsson、Senger 和 Sheffer 最近提出了 Erdős 单位距离问题的以下概括。对于k距离的固定序列δ = ( δ ₁ , …, δ _k ) ，一个 ( k + 1)-元组 ( p ₁ , …, p _{k +1} ) 的不同点在 ℝ ^d中称为k 链如果 ∥ p _j − p _{j +1} ∥ = δ _j对于每个 1 ≤ j ≤ k。C ^d _k的最大数是多少 ( n ) k链在 ℝ ^d中的一组n点中？改进 Palsson、Senger 和 Sheffer 的结果，我们基本上确定了平面情况下所有k的最大值。仅对于k ≡ 1 (mod 3)，答案取决于一组n个点中的最大单位距离数。我们还对第 3 维中的偶数k获得了几乎清晰的结果，并提出了进一步的概括。