Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have $\chi^{-1}(\{\beta\}) \in \mathrm{Ind}(G)$ for all $\beta \in \kappa$.

Let $\mathrm{Max}(\mathrm{Ind}(G))$ be the set of maximal independent sets of $G$ and suppose $\kappa_0$ is the smallest cardinal such that there is a coloring map $\chi: G\to \kappa_0$.

**Question**: Is it possible that for all coloring maps $\chi: G\to \kappa_0$ we have $\{\chi^{-1}(\{\beta\}) : \beta\in\kappa_0\} \cap \mathrm{Max}(\mathrm{Ind}(G)) = \emptyset$?