The injection-multiplicity Ramsey conjecture

Let $H$ be a graph, and let $G$ be a graph of order $n$ with average degree $\overline d(G)$. Write $v(H)$ and $e(H)$ for the numbers of vertices and edges of $H$, and let $\operatorname{inj}(H,G)$ denote the number of injective homomorphisms from $H$ to $G$. Injection-multiplicity Ramsey conjecture. There exists a constant $h=h(H)<1$ such that, for any such $G$ with

$$\overline d(G)\in n^{h+o(1)},$$ $$\operatorname{inj}(H,G)\in n^{v(H)-(1-h)e(H)-\Omega(1)}$$

implies

$$\alpha(G)\in n^{1-h+\Omega(1)}.$$

This conjecture asks whether sufficiently few injective copies of $H$ force an independence number larger than the scale suggested by the average degree. It is introduced as being informed by the preceding results, with no resolution evidence supplied in the text.