On the exact maximum induced density of almost all graphs and their inducibility

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = \Pi_{i}^h a_i$ where $\sum_{i=1}^h a_i = n$ and the $a_i$ are as equal as possible. Let $g(n,h) = f(n,h) + \sum_{i=1}^h g(a_i,h)$. It is proved that for almost all graphs $H$ on $h$ vertices it holds that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. More precisely, we define an explicit graph property ${\cal P}_h$ which, when satisfied by $H$, guarantees that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. It is proved, in particular, that a random graph on $h$ vertices satisfies ${\cal P}_h$ with probability $1-o_h(1)$. Furthermore, all extremal $n$-vertex graphs yielding $i_H(n)$ in the aforementioned range are determined. We also prove a stability result. For $H \in {\cal P}_h$ and a graph $G$ with $n \le 2^{\sqrt{h}}$ vertices satisfying $i_H(G) \ge f(n,h)$, it must be that $G$ is obtained from a balanced blowup of $H$ by adding some edges inside the blowup parts. The {\em inducibility} of $H$ is $i_H = \lim_{n \rightarrow \infty} i_H(n)/\binom{n}{h}$. It is known that $i_H \ge h!/(h^h-h)$ for all graphs $H$ and that a random graph $H$ satisfies almost surely that $i_H \le h^{3\log h}h!/(h^h-h)$. We improve upon this upper bound almost matching the lower bound. It is shown that a graph $H$ which satisfies ${\cal P}_h$ has $i_H =(1+O(h^{-h^{1/3}}))h!/(h^h-h)$.