The countably chromatic quantifier $Q^{\mathrm{Chr}}_{\omega}$ can be formulated by taking binary $S$ and the class $\begin{align} \mathfrak{R}= & \{ \mathfrak{A} \mid (A,S^{\mathfrak{A}}) \text{ is a countably chromatic graph} \} \\ = & \{ \mathfrak{A} \mid \exists c:S^{\mathfrak{A}}\to \omega \ \forall a,b\in A[S^{\mathfrak{A}}(a,b)\to c(a)\ne c(b)] \} \end{align} $