On the number of discrete chains

Determining the maximum number of unit distances that can be spanned by points in the plane is a difficult problem, which is wide open. The following more general question was recently considered by Eyvindur Ari Palsson, Steven Senger, and Adam Sheffer. For given distances $t_1, \ldots, t_k$ a $(k+1)$-tuple $(p_1, \ldots, p_{k+1})$ is called a $k$-chain if $||x_i-x_{i+1}||=t_i$ for $i=1, \ldots, k$. What is the maximum possible number of $k$-chains that can be spanned by a set of $n$ points in the plane? Improving the result of Palsson, Senger and Sheffer, we determine this maximum up to a small error term (which, for $k=1 \mod 3$ involves the maximum number of unit distances). We also consider some generalisations, and the analogous question in $\mathbb R^3$.

Joint work with Andrey Kupvaskii.