  The following generalisation of the Erd\H{o}s unit distance problem was
recently suggested by Palsson, Senger and Sheffer. Given $k$ positive real
numbers $\delta_1,\dots,\delta_k$, a $(k+1)$-tuple $(p_1,\dots,p_{k+1})$ in
$\mathbb{R}^d$ is called a $(\delta,k)$-chain if $\|p_j-p_{j+1}\| = \delta_j$
for every $1\leq j \leq k$. What is the maximum number $C_k^d(n)$ of
$(k,\delta)$-chains in a set of $n$ points in $\mathbb{R}^d$, where the maximum
is taken over all $\delta$? Improving the results of Palsson, Senger and
Sheffer, we essentially determine this maximum for all $k$ in the planar case.
error term It is only for $k\equiv 1$ (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 $3$ dimension.
