1) As you say in the question, Ramsey numbers, both the (easy) upper bound - an elegant example of the power of combinatorics - and the lower bound, one of the rare proofs that is easy, short, and completely surpising.
2) Turan's theorem about the size of independent sets in graphs. Again, an easy application of the probabilistic method (though you don't even need the language of probability to state it). It's also interesting that this is an example, unlike Ramsey numbers, where the 'easy' proof actually gives the best possible bound (for general graphs).
Of course given the wide variety of problems graphs are applicable to, it's useful to be able to detect structured subgraphs, either complete graphs or independent sets. These are also good examples of non-constructive proofs.