[1] Informally, it states that all NP-complete languages look alike, in the sense that they can be related to each other by polynomial time isomorphisms.
[1] A formal language L is paddable if there is a polynomial time function f(x,y), with a polynomial time inverse, such that for every x and every y, the string x belongs to L if and only if f(x,y) belongs to L. That is, it is possible to pad the input x with irrelevant information y, in an invertible way, without changing its membership in the language.
Berman and Hartmanis proved that all pairs of paddable NP-complete languages are p-isomorphic.
[8] And Fenner, Fortnow & Kurtz (1992) found an oracle machine model in which the analogue to the isomorphism conjecture is true.
[9] Evidence against the conjecture was provided by Joseph & Young (1985) and Kurtz, Mahaney & Royer (1995).