Largest sparse subgraphs of random graphs

摘要

For the Erd?s–Rényi random graph G_(n,p), we give a precise asymptotic formula for the size at (Gn,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.