| Summary: | For the Erdős–Rényi random graph G<sub>n,p</sub>, we give a precise asymptotic formula for the size <em>â<sub>1</sub></em><em>(G<sub>n,p</sub></em>) of a largest vertex subset in G<sub>n,p</sub> 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.
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