Approximating the clique size of the graph
It's not possible. The reason is that, to distinguish an $n$-vertex graph with no edges (clique size 1) from a graph with a single randomly-chosen edge (clique size 2) requires $\Theta(n^2)$ queries. So it's not possible to get sublinear queries for $\rho=2$ and $\epsilon&lt;\frac12$, but your question asks for a parameterization that remains sublinear for all $\rho$ and all $\epsilon&gt;0$.