Largest sparse subgraphs of random graphs - Fountoulakis, N; Kang, R; McDiarmid, C (2014)
For the Erdodouble acutes-Rényi random graph G n,p, we give a precise asymptotic formula for the size α̂t(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. © 2013 Elsevier Ltd.