Applications of compactness theorem:
2-colorability of graphs
 
Goal: Given a graph G=(V,E)  it is 2-colorable iff every finite subset of G is 2-colorable. 

Proof outline:  
Given a graph G=(V,E) construct a set \Sigma of wffs such that
G is 2-colorable  
iff  (step 1) 
\Sigma is satisfiable. 
iff  
(by CT) \Sigma is finitely satisfiable
iff (step 2) 
Each finite subset of G is 2-colorable.