Erdős problem 1007

The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\mathbb{R}^n$ such that every edge of $G$ is a unit line segment.

What is the smallest number of edges in a graph with dimension $4$?

The smallest number of edges is $9$, achieved solely by $K_{3,3}$, proved by House [Ho13]. An alternative proof was given by Chaffee and Noble [ChNo16], who also prove that the smallest number of edges in a graph of dimension $5$ is $15$ (achieved by $K_6$ and $K_{1,3,3}$).