I've encountered an interesting problem but can solve it only partially:



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> 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.
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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.



Random graphs nonisomorphic to unit distance graphs
The almost sure asymptotic chromatic number of $G$ goes to $\infty$ with $c$, see for example the precise result by Achlioptas and Naor in Annals of Math. 2005.


The chromatic number of a unit-distance graph (and in fact of the whole plane) is bounded above by $7$, see e.g. the math coloring book by Soifer (this is simple: one colors an hexagonal tiling of carefully chosen side length).


These two facts end the proof of your problem.


