TITLE: Equivalent forms of the P vs. NP problem
QUESTION [14 upvotes]: Many things in math can be formulated quite differently; see the list of statements equivalent to RH here, for example, with RH formulated as a bound on lcm of consecutive integers, as an integral equality, etc.
I wonder about equivalent formulations of the N vs. NP problem. Formulations that are very much different from the questions such "Is TSP in P?", formulation that may seem unrelated to complexity theory.

REPLY [2 votes]: The P vs NP problem can be formulated in terms of incomplete sets in NP. Ladner theorem can be stated as:
$P \ne NP$ if and only if there is an incomplete set in NP.
Incomplete set is a set that is not complete for $NP$ under many-one polynomial time reductions (Karp reductions).
Another formulation in terms of sparse sets is Mahaney's Theorem:
There is no sparse NP-complete set if and only if $P \ne NP$
(under Karp reduction).
Complexity Theory and Cryptology: An Introduction to Cryptocomplexity By Jörg Rothe, page 106