I am interested whether there exists a certain generalization of so-called compactness argument in graph theory.
First let me define what I mean by "compactness argument"
Let $G = (V, E)$ be an infinite graph. Let $F$ be the collection of all it's $k$-colorings (we consider any colorings here, not just proper colorings). Now we want to specify for each finite subgraph of $G$ a collection of colorings we call "interesting colorings". Let $F'(G')$ be a collection of interesting colorings of a finite $G' \subset G$. We demand from $F'$ to have two properties
- $F'(G')$ is non empty for any $G'$, in other words every finite subgraph admits an interesting coloring
- If $G'' \subset G'$ and $f \in G'$ then $f$ restricted to $G''$ is in $F'(G')$.
If we have such a setting then one can prove that there exists a coloring $f$ of the whole $G$ such that for any finite subgraph $G'$, $f$ restricted to $G'$ is in $F'(G')$.
The proof of this statement can be done using Tychonoff's theorem using compactness of space $\{1, \ldots, k\}^{|G|}$ I am interested in a question whether a similar theorem remains true if we replace the word "finite" with the word "countable" in the theorem's text.