I've encountered an interesting problem but can solve it only partially:
Prove that random graph $G\sim G\left(n,\frac cn\right)$, $c=const$, almost surely is isomorphic to some unit distance graph on a plane if $c$ is sufficiently small and almost surely won't be ismorphic to any unit distance graph on a plane if $c$ is sufficiently large.
I can prove only the first part (for $c<1$, connected components of $G$ will almost surely contain no more than 1 cycle, and it's easy to show that such $G$ can be represented as a unit distance graph).
Some precisions: the model for random graphs is the one of Erdös and Renyi ($G$ has $n$ vertices and each edge is present with probability $c/n$), and ``almost surely'' means that the probability of the event goes to $1$ when $n\to\infty$.
A unit distance graph is a graph that can be represented by point in the plane, with two points joined by an edge if and only if their are at unit distance one from the other.