A converse of the Erdős-De Bruijn Theorem?

For the chromatic number $$\chi(G)$$ of a simple, undirected graph, there is a "compactness" theorem by Erdős and De Bruijn 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?

• Isn't $(S)$ the usual statement of the Erdös-De Bruijn theorem? From the second line in the wikipedia link: "It states that, when all finite subgraphs can be colored with $c$ colors, the same is true for the whole graph" Commented Mar 14, 2020 at 8:22
• But even if you take the statement in the question, $(S)$ is an immediate corollary. Suppose $\chi(G_0)=k$ for every finite $G_0\subset G$. Obviously $\chi(G)\geq k$. But we can't have $\chi(G)>k$ because that would be witnessed by a finite subgraph. Commented Mar 14, 2020 at 8:26
• This can be proven by applying compactness for 1st order logic, right? Commented Mar 14, 2020 at 8:59
• @Monroe that's right Commented Mar 14, 2020 at 9:31