Largest sparse subgraphs of random graphs

For the Erdős–Rényi random graph Gn,p, we give a precise asymptotic formula for the size â1(Gn,p) of a largest vertex subset in Gn,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