Q1. How dense can a nearly-unit-distance graph be?
Let points sit in $\mathbb{R}^2$.
A unit-distance graph UDG "connect[s] two points by an edge whenever the distance between the two points is exactly $1$."
An upper bound on the density (number of pairs at distance $1$) is $O(n^{4/3})$.
Let $0 < \epsilon \ll 1$. I would like to loosen the "exactly" criterion and allow edges in an ~UGD that are within $\epsilon > 0$ of $1$. Then the density is the number of pairs within distance $1 + \epsilon$.
Q2. Does this immediately lead to $\Omega(n^2)$ behavior?
I.e.,
Q3. Is there a set of $n$ points such that $\Omega(n^2)$ of them (some fraction of $n^2$ of them) are within $1+\epsilon$ of one another?
Let half the points be tightly clustered about $(0,0)$ and the other half around $(1,0)$, i.e. within $\epsilon/2$ of the respective endpoints.
Then we have $\lfloor n/2 \rfloor \cdot \lceil n/2 \rceil$ edges of length between $1-\epsilon$ and $1+\epsilon$. Thus asymptotically we can arrange $n^2/4 + O(1)$ such edges using $n$ points in the plane.
This is not the best possible fraction of edges. We could get $n^2/3 + O(n)$ edges by clustering points as equally as possible about the three vertices of a unit equilateral triangle.