We show that, given an infinite cardinal $\mu$, a graph has colouring number at most $\mu$ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a well-ordering of its vertices that simultaneously witnesses its colouring number and its cardinality.

中文翻译：

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无限图的着色数

我们证明，给定无限基数 $\mu$，当且仅当它不包含两种类型的子图时，图最多具有 $\mu$ 的着色数。我们还表明，每个具有无限着色数的图都有其顶点的有序性，同时见证了它的着色数和基数。