We study the question of whether the class DisNP of
disjoint pairs (A, B) of NP-sets contains a complete pair.
The question relates to the question of whether optimal
proof systems exist, and we relate it to the previously
studied question of whether there exists a disjoint pair
of NP-sets that is NP-hard. We show under reasonable
hypotheses that nonsymmetric disjoint NP-pairs exist,
which provides additional evidence for the existence of
P-inseparable disjoint NP-pairs.

We construct an oracle relative to which the class of
disjoint NP-pairs does not have a complete pair,
an oracle relative to which optimal proof systems exist, hence
complete pairs exist, but no pair is NP-hard, and an oracle
relative to which complete pairs exist, but optimal proof systems do
not exist.