Induced copy density and homogeneous restriction parameter

Let $G$ be a nonempty finite simple graph, $n=\mathrm{\mid }V(G)\mathrm{\mid }$, and $H$ a finite simple graph with $h=\mathrm{\mid }V(H)\mathrm{\mid }$. Define $d_{\mathrm{i}\mathrm{n}\mathrm{d}}(H,G)={\mathrm{ind}}_H(G)\mathrm{/}{n}^h$, using the labelled induced embeddings of The induced-embedding count ${\mathrm{ind}}_H(G)$ and Induced embeddings and induced copies of a graph.

For $a,b\ge 0$, put ${\rho }_G(a,b)=\max \{\frac{\mathrm{\mid }S\mathrm{\mid }}{n}:\mathrm{\varnothing }\mathrm{\ne }S\subseteq V(G),e(G[S])\le a\left(\genfrac{}{}{0px}{}{\mathrm{\mid }S\mathrm{\mid }}{2}\right)\text{ or }e(\bar{G}[S])\le b\left(\genfrac{}{}{0px}{}{\mathrm{\mid }S\mathrm{\mid }}{2}\right)\}\mathrm{.}$ Here $e(G[S])$ counts unordered edges. For disjoint sets, $e_G(A,B)$ counts cross edges as in Edge counts and densities between nonempty vertex sets. We use the quotient $e(G[S])\mathrm{/}\left(\genfrac{}{}{0px}{}{\mathrm{\mid }S\mathrm{\mid }}{2}\right)$ only for $\mathrm{\mid }S\mathrm{\mid }\ge 2$.

There are finitely many subsets by $\mid \mathcal{P}(A)\mid =2^{\mid A\mid }$ for finite $A$. A singleton has no edges and $\left(\genfrac{}{}{0px}{}{1}{2}\right)=0$ by A finite set with $n$ elements has exactly $\left(\genfrac{}{}{0px}{}{n}{2}\right)$ two-element subsets, and $2\left(\genfrac{}{}{0px}{}{n}{2}\right)=n(n-1)$, so the family in the maximum is nonempty. Comparing a finite list of its real values gives an attained maximum, with $1\mathrm{/}n\le {\rho }_G(a,b)\le 1$. If $a\ge 1$ or $b\ge 1$, the full vertex set qualifies and ${\rho }_G(a,b)=1$.

For the null pattern, the unique empty map is an induced embedding, so ${\mathrm{ind}}_{\mathrm{\varnothing }}(G)=1$ and $d_{\mathrm{i}\mathrm{n}\mathrm{d}}(\mathrm{\varnothing },G)=1$. For a one-vertex pattern the count is $n$.

Proof/convention locator: Bucic, Nguyen, Scott and Seymour, Induced subgraph density I, Sections 2 and 5, before 5.2 and its beta_s definition.