For the chromatic number $\chi(G)$ of a simple, undirected graph, there is a ["compactness" theorem by Erdős and De Bruijn](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)) stating that if an infinite graph $G$ has finite chromatic number, then there is a finite subgraph $G\_0\subseteq G$ such that $\chi(G\_0) = \chi(G)$.


A converse statement would be



> 
> $(\text{S})$ Let $k>0$ be an integer. Whenever a graph $G$ has the property that every finite subgraph $G\_0$ of $G$ can be colored with $k$ colors, then $G$ can be colored with $k$ colors.
> 
> 
> 


Is $(\text{S})$ true?



A converse of the Erdős-De Bruijn Theorem?
Apparently yes. By googling, I found the assertion in [this 1951 paper of de Bruijn and Erdős](http://combinatorica.hu/~p_erdos/1951-01.pdf), and the first page contains several further references.


