Largest sparse subgraphs of random graphs

For the Erd\H{o}s-R\'enyi random graph $G_{n,p}$, we consider the order of a largest vertex subset that induces a subgraph with average degree at most $t$. For the case when both $p$ and $t$ are fixed, this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.