Let $G=(V, E)$ be an arbitrary graph with $n := |V|$ vertices and let $r \leq n$ be a parameter. Let $U \subseteq V$ be a subset of size $r$, chosen uniformly at random from the set of all subsets of $V$ that have size $r$. Finally, let $X$ be the number of edges inside the induced subgraph $G[U]$ (this is the subgraph of $G$ including all the edges with both endpoints in $U$). How concentrated is random variable $X$ around its expectation?
One can use Chernoff-Hoeffding type bounds to first bound the maximum degree $\Delta$ of $G[U]$ and give an upper bound of $\Delta \times r$ on $X$. But this is a rather weak bound. Any other suggestions?