Let $G = (V,E)$ be a simple, undirected graph. For $v\in V$ we let $N(v) = \{w \in V: \{v,w\} \in E\}$.

We define the *coloring number* $\text{Col}(G)$ of the graph $G$ to be the smallest cardinal $\kappa$ such that there is a well-ordering $\leq_{\text{well}}$ on $V$ such that for every vertex $v\in V$ we have $$|N(v) \cap \{w\in V: w \leq_{\text{well}} v\}|< \kappa.$$

**Question.** Is there an infinite graph $G = (V,E)$ such that $\chi(G)$ is finite but $\text{Col}(G)$ is infinite?