The Baker-Gill-Solovay result showed that the P = NP question does not relativize, in the sense that no relativizing proof (insensitive to the presence of an oracle) can possibly settle the P = NP question.
My question is: Is there a similar result for the question, "Does there exist a PH-complete problem?" An answer in the negative to this question would imply P != NP; an answer in the affirmative would be unlikely but interesting because it would mean that PH collapses to some level.
I'm not sure, but I suspect that a TQBF oracle would lead PH to be equal to PSPACE, and thus to have a complete problem. In addition to being uncertain regarding this, I am curious as to whether or not there is an oracle relative to which PH provably does not have a complete problem.