On the number of discrete chains
We study a generalization of the Erdos unit distance problem to chains of k distances. Given P, a set of n points, and a sequence of distances (δ1, . . . , δk), we study the maximum possible number of tuples of distinct points (p1, . . . , pk+1) ϵ Pk+1 satisfying |pjpj+1| = δj for every 1 ≤ j ≤ k. We study the problem in R2 and in R3, and derive upper and lower bounds for this family of problems.
Palsson, Eyvindur, Steven Senger, and Adam Sheffer. "On the number of discrete chains." Proceedings of the American Mathematical Society 149, no. 12 (2021): 5347-5358.
Proceedings of the American Mathematical Society