Recently I saw some graph theory proofs works like the following:
Theorem: If A then B.
Proof: Assume G be a counterexample of A, then saturate the graph(as in add edges) until adding any edge will violate A. Let the saturated graph be G'. Then use G' to derive a contradiction.
Why don't we just chose a saturated graph instead of spending a few sentence to pick any counterexample and then build a saturated example?
Note: A a saturated counterexample always exists if there is a counterexample. For any finite non-empty set, give an partial order, then there exist a maximal element. One can use this to prove any finite non-empty set of graphs A, there has to be a saturated graph(as in adding one more edge, the graph is no longer in A).