Let $t \gt 0$ be an integer and $G$ is a simple graph with $\chi_f(G) = t$. Then $t$= inf $\{ \frac{n}{k}| G \rightarrow KG(n,k)\}$ where $KG(n,k)$ is the Kneser graph.

Does there exist a $k$ such that $G \rightarrow KG(tk,k)$? If so, what is the complexity of finding such a $k$?