On the sizes of large subgraphs of the binomial random graph

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 ω_n→∞, a.a.s. it is smaller than ω_nn/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)).