Mahaney's theorem explained

Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that if any sparse language is NP-complete, then P = NP. Also, if any sparse language is NP-complete with respect to Turing reductions, then the polynomial-time hierarchy collapses to

	P
\Delta
	2

.^[1]

Mahaney's argument does not actually require the sparse language to be in NP, so there is a sparse NP-hard set if and only if P = NP. This is because the existence of an NP-hard sparse set implies the existence of an NP-complete sparse set.^[2]

Notes and References

Mahaney. Stephen R.. Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences. October 1982. 25. 2. 130–143. 10.1016/0022-0000(82)90002-2. 1813/6257. free.
Book: Structural Complexity II. Balcázar. José Luis. Díaz. Josep. Gabarró. Joaquim. 1990. 3-540-52079-1. Springer. 130–131.