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.

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Elsevier B.V.
J. Nešetřil (Jaroslav) , E. Győri , A. Sali
Electronic Notes in Discrete Mathematics
European Conference on Combinatorics, Graph Theory and Applications

Fountoulakis, N., Kang, R., & McDiarmid, C. (2011). Largest sparse subgraphs of random graphs. In J. Nešetřil, E. Győri, & A. Sali (Eds.), Electronic Notes in Discrete Mathematics. Elsevier B.V.