Exercise. Let $n>2$ be a natural number. Define the simple graph $G_n=(V,E)$ as follows: $$V=\{A\subset\{1,2,...,n\}:|A|=2\}\ , \{A,B\}\in E\iff A\cap B=\emptyset. $$ For which values of $n$ is $G_n=(V,E)$ connected?
Attempt. I could not solve the problem satisfying the above conditions, so I tried to simplify the problem and find out for which values of $n$ the graph $G_n=(V, E)$ is a tree, a minimally connected graph. But I still have not succeeded in finding some bounds for $n$. I cannot understand how can I make sure that $G_n=(V, E)$ is a connected graph and find the corresponding bounds of $n$.
Any help would be appreciated.
The graph is connected for $n \geq 5$. As others have said in the comments, when $n = 3$ the graph has no edges and when $n = 4$ the graph has 6 vertices, but only two edges.
Now assume $n \geq 5$ and take two non-adjacent vertices. Without loss of generality, we may assume the vertices are $\{1,2\}$ and $\{2,3\}$. A path between these two vertices is given by $$\{1,2\} \rightarrow \{4,5\} \rightarrow \{2,3\}.$$
EDIT: For $n = 4$, there are three edges. Thank you to @AtticusStonestrom for catching my mistake.