The Ramsey-Sidorenko conjecture for homomorphism-multiplicity bounds

Let $H$ be a graph, and let $G$ be a graph of order $n$. Write $v(H)$ and $e(H)$ for the numbers of vertices and edges of $H$, $\hom(H,G)$ for the number of homomorphisms from $H$ to $G$, and $\alpha(G)$ for the independence number of $G$. Ramsey-Sidorenko conjecture. There exist constants $a,C_H>0$ such that

$$\alpha(G)\geq C_H\min\left\{n^{a},\left(\frac{n^{v(H)}}{\hom(H,G)}\right)^{1/e(H)}\right\}.$$

This proposes a homomorphism-multiplicity analogue of Ramsey bounds: controlling the number of copies of $H$ should force a large independent set. The statement is presented as plausible and is informed by results including the Bohman–Mubayi theorem for cliques; its status is otherwise not resolved in the supplied text.