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Here's a question. Have there been any theoretical results showing that if $P \neq NP$, there must exist some problems in $NP$ which are not $NP$-complete and which are not in $P$ either? Just curious because I've never seen this question addressed.

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Yes, this is known as Ladner's theorem, proved by Richard Ladner in 1975. The class of such problems are called NP-intermediate. Here's one proof that does "cut and paste" between some problem in P and some NP-complete problem, here are two, and there are others.

There are also problems that are already believed to be neither NP-complete nor in P, like integer factorisation and graph isomorphism, as well as problems known to be in NP ∩ coNP but not known to be in P.

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In this connection I can't resist linking to Ryan Williams' excellent explanation of Ladner's theorem on MO: mathoverflow.net/questions/9221/… –  Qiaochu Yuan Aug 14 '10 at 7:48
    
@Qiaochu: Thanks, describing it as "cut and paste" seems so boring now. :-) –  ShreevatsaR Aug 14 '10 at 7:55

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