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 $e(k)=p{k\choose 2}+O(k)$, 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 $C>0$ and every $\omega_n\to\infty$, a.a.s.~the size of the concentration set belongs to $(C\sqrt{n/\ln n},\omega_n\sqrt{n/\ln n})$. Second, given $k>\varepsilon n$, what is the maximum $\mu$ such that a.a.s.~the set of sizes of $k$-vertex subgraphs of $G(n,p)$ contains a full interval of length $\mu$? The answer is $\mu=\Theta\left(\sqrt{(n-k)n\ln{n\choose k}}\right)$.

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