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 (delta(1), ..., delta(k)), we study the maximum possible number of tuples of distinct points (p(1), ..., Pk+1) is an element of Pk + 1 satisfying vertical bar p(j)p(j+1)vertical bar= delta(j) for every 1 <= j <= k. We study the problem in R-2 and in R-3 , and derive upper and lower bounds for this family of problems.