The class of pairs (A,B), where A and B are NP problems whose sets of "yes" instances are nonempty and disjoint.
If there exists an optimal propositional proof system, then DisNP has a complete pair [Raz94]. On the other hand, there exists an oracle relative to which DisNP does not have a complete pair [GSS+03].
If P does not equal UP, then DisNP contains pairs not separated by any set in P [GS88]. On the other hand, there exists an oracle relative to which P does not equal NP but still DisNP does not contain any P-inseparable pairs [HS92].
No class.