On the sizes of large subgraphs of the binomial random graph

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We consider the binomial random graph G(n,p), where p is a constant, and answer the following two questions.First, given e(k)=p(k2)+O(k), what is the maximum k such that a.a.s. the binomial random graph G(n,p) has an induced subgraph with k vertices and e(k) edges? We prove that this maximum is not concentrated in any finite set (in contrast to the case of a small e(k)). Moreover, for every constant C>0, with probability bounded away from 0, the size of the concentration set is bigger than Cn/ln⁡n, and, for every ωn→∞, a.a.s. it is smaller than ωnn/ln⁡n.Second, given k>εn, what is the maximum μ such that a.a.s. the set of sizes of k-vertex subgraphs of G(n,p) contains a full interval of length μ? The answer is μ=Θ((n−k)nln⁡(nk)).