I think the meaning of the term "exist" needs to be clarified. All of the examples you describe except the Ramsey-theoretic one depend on axioms independent of ZF (e.g. the ultrafilter lemma). On the other hand, the <a href="http://en.wikipedia.org/wiki/Probabilistic_method">probabilistic method</a> can prove, _in ZF_, that plenty of objects exist (e.g. efficient sphere packings, families of graphs realizing bounds on the Ramsey numbers) for which we do not have deterministic constructions. I assume this is what Harrison is referring to (the use of the probabilistic method in Ramsey theory).