The bridge: Problem 97 implies the 3n bound and Problem 96

Assume that every nonempty finite set $C$ of points in the real Euclidean plane whose points all lie outside the real convex hull of the remaining points fails to have the following property: for every $p\in C$ there is a positive real radius $r>0$ such that $\{q\in C : \operatorname{dist}(p,q)=r\}$ has cardinality at least $4$. Then both of the following hold: for every finite set $A$ of plane points satisfying the same convex-hull condition, including the possibility that $A$ is empty, the number $\operatorname{unitDistancePairsCount}(A)$ of unordered two-element pairs from $A$ counted once, where a pair $\{p,q\}$ is counted exactly when $0<\operatorname{dist}(p,q)$ and $\operatorname{dist}(p,q)=1$, is at most $3|A|$; and the real-valued function sending $n$ to the natural-number supremum of these unit-distance counts over finite convex-independent point sets of cardinality $n$ grows as $O(n)$ along natural numbers tending to infinity.