# On the Number of Discrete Chains

## TL;DR

This paper extends Erdős's unit distances problem to chains of multiple distances, providing bounds on the maximum number of such chains in planar and three-dimensional point sets.

## Contribution

It introduces a generalized problem of distance chains and derives new upper and lower bounds in two and three dimensions.

## Key findings

- Derived bounds for chains of distances in 2 and 3.
- Extended Erd
53s's problem to multiple distances.
- Provided theoretical limits for maximum chain counts.

## Abstract

We study a generalization of Erd\H os's unit distances problem to chains of $k$ distances. Given $\mathcal P,$ a set of $n$ points, and a sequence of distances $(\delta_1,\ldots,\delta_k)$, we study the maximum possible number of tuples of distinct points $(p_1,\ldots,p_{k+1})\in \mathcal P^{k+1}$ satisfying $|p_j p_{j+1}|=\delta_j$ for every $1\leq j \leq k$. We study the problem in $\mathbb R^2$ and in $\mathbb R^3$, and derive upper and lower bounds for this family of problems.