We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of S21 exists in which NP is not in the second level of the linear hierarchy; and that a model of S21 exists in which the polynomial hierarchy collapses to the linear hierarchy.
Our methods are model-theoretic. We use the assumption about factoring to get a model in which the weak pigeonhole principle fails in a certain way, and then work with this failure to obtain our results.
As a corollary of one of the proofs, we also show that in S21 the failure of WPHP (for Σb1 definable relations) implies that the strict version of PH does not collapse to a finite level.
"The polynomial and linear hierarchies in models where the weak pigeonhole principle fails." J. Symbolic Logic 73 (2) 578 - 592, June 2008. https://doi.org/10.2178/jsl/1208359061