Coloring theorem implies the compactness theorem Compactness theorem: Let $\Sigma$ be a set of Formulas. If every finite subset of $\Sigma$ has a realization then $\Sigma$ has a realization too.
Definition[Realisation]: Let $\Sigma$ be a set of formulas, $S_\Sigma\subset\mathcal{P}$ be a the set of sentences which appears in $\Sigma$. $f^*:\Phi\to \{0,1\}$ where $\Phi$ is a set of formulas and $f^*(\phi\wedge\psi)=f^*(\phi)f^*(\psi)$, $f^*(\phi\lor\psi)=\max\{\phi,\psi\}$, $f^*(\lnot\phi)=1-f^*(\phi)$ and for all $\phi\in \Sigma$ we have $f^*(\phi)=1$. $f:S_\Sigma\to \{0,1\},p\mapsto f^*(p)$ is called a realisation of $\Sigma$.
Coloring theorem: Every graph which has every finite subgraph which is 3-colorable is 3-colorable.
How does the coloring theorem implies the compactness theorem?
 A: This is an interesting question, since usually it is the converse that is proven, namely that the compactness theorem (for propositional logic) implies the 3-colouring theorem.
To go the other way, we need to construct a graph from the set of formulae. Start with a triangle with vertices $S_0,S_1,S_2$. We can assume that any $3$-colouring will colour them $0,1,2$ respectively. Add a vertex for each variable and add an edge from it to $S_2$. Then every variable will be coloured $0$ or $1$, which of course we will interpret as truth values. It is easy to construct gadgets using $S_{0..2}$ to compute the value of any boolean operation on the values of two given vertices. Thus for each formula we can add a finite subgraph of which one vertex will be coloured according to its truth value given that the variables' vertices have been coloured according to the truth assignment. Merge that vertex with $S_1$. This forces the colouring to be possible only if the truth assignment satisfies the formula. Now observe that any finite set of formulae is captured by a finite subgraph, so the $3$-colouring theorem can be applied to yield the compactness theorem.
A: Another compactness related concept is the one of Boolean prime ideal principle, BPI, which asserts that every collection of elements on an arbitrary Boolean algebra with finite intersection property (every two elements have nonzero meet), FIP, can be extended to an ultrafilter.
BPI is known to be equivalent to the coloring principle: see H.Lauchli's 1971 paper Coloring infinite graphs and the Boolean prime ideal theorem.
We can then transfer this question to an algebraic one using Lindenbaum-Tarski algebras. Let $F$ denote the set of all formluas of classical logic and define  its quotient $F/_\cong$, where $\varphi\cong\psi$ iff these two formulas are equiprovable in classical logic (equivalently, they have the same value in every realisation). So the elements of $F/_\cong$ have the form $[\varphi]=\{\psi\, |\,\psi\cong\varphi \}$. We can then naturally equip this set with boolean operations:  $[\varphi]\wedge [\psi]=[\varphi\wedge\psi]$ and similarly for other connectives. Note that its zero has a form $[\bot]$, the equivalence class of contradictory formulas, e.q. $[\psi\wedge\neg\psi]=[\bot]$. $F/_\cong$ with these operations then from a Boolean algebra, which we call a Lindenbaum-Tarski algebra, lets denote it $\mathrm{LT}$.
For a set of formulas $\Sigma$ define $[\Sigma]:=\{[\psi]\, |\, \psi\in \Sigma\}$. Quite easily one can show the following two equivalences:


*

*$\Sigma$ is finitely realisable.

*$[\Sigma]$ has FIP in $\mathrm{LT}$.


(See that if $\psi_1\dots\psi_n\in\Sigma$ then there is a realiastion $v$ such that $v(\psi_i)=1$, thus $\bigwedge \psi_i\ncong \bot$ and consequently $\bigwedge[\psi_i]=[\bigwedge \psi_i]\neq [\bot]=0$). The second equivalence: 


*

*$\Sigma$ is maximal realisable set of formulae.

*$[\Sigma]$ is an ultrafilter in $\mathrm{LT}$.


The witnessing realisation is given by $f(v)=1$ iff [v] is in the ultrafilter $[\Sigma]$. Thus BPI allows us to extend $\Sigma$ to a bigger set of formulas which is realisable (of course, $\Sigma$ itself will be realisable).
