Oracle-dependent properties of the lattice of NP sets

AbstractWe consider under the assumption P ≠ NP questions concerning the structure of the lattice of NP sets together with the sublattice P. We show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize. The two questions which we consider are whether every infinite NP set contains an infinite P subset and whether there exists an NP-simple set. We construct several oracles, all of which make P ≠ NP, and which in addition make the above-mentioned statements either true or false. In particular we give a positive answer to the question, raised by Bennett and Gill (1981), whether an oracle B exists making PB ≠ NPB and such that every infinite set in NPB has an infinite subset in PB. The constructions of the oracles are finite injury priority arguments