Largest sparse subgraphs of random graphs

For the Erdős–Rényi random graph G n , p , we give a precise asymptotic formula for the size α ˆ t ( G n , p ) of a largest vertex subset in G n , p that induces a subgraph with average degree at most t , provided that p = p ( n ) is not too small and t = t ( n ) is not too large. In the case of fixed t and p , we find that 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.