Avoiding Distances

Given a finite planar point set $X=\{x_1,\ldots,x_n\}$, let $a_1<a_2< \cdots < a_s$ denote the $s \leq {n \choose 2}$ different (Euclidean) distances between points in $X$. The aim is to retain as many of the points as possible such that certain imposed inter-pair distances are avoided. For example,