Can NP-Intermediate exist if P = NP?

Question

My understanding is that Ladner's theorem is basically this:

P != NP implies that there exists a set NPI where NPI is not in P and NPI is not NP-complete

What happens to this theorem if we assume that P = NP rather than P != NP? We know that if NP Intermediate doesn't exist, then P = NP. But can NP Intermediate exist if P = NP?

Answer 1

You missed one property of NPI: Every element of NPI is in NP (but not in P). This is clearly impossible if P=NP, so if P=NP, NPI must be empty.

Answer 2

If P=NP, then NPI cannot exist assuming that it is a subset of NP, as all of NP is in P and thus the "not in P" part of the definition of NPI would not hold for any problem. So the class NPI would be empty in that case.

Answer 3

NPI must imply that it is in NP, but that it is not NP-complete.

If P = NP, then all problems in P and NP will be NP-complete, because any problem will be reducible to another one in polynomial time (∅ and Σ* cannot be NP-complete, because we can't map an arbitrary problem to either of them - we won't have anything to map to for the positive/negative case. However, since they are in P, we don't care about them for the purpose of this question.)

Since all problems in NP are NP-complete, NPI cannot exist.