A set is sparse if it contains polynomially bounded number of strings of any given string length $n$ otherwise it is dense. All known NP-complete sets are dense. It was proven that P=NP if and only if there is a sparse NP-complete set (under Karp reduction).
I would like to find the density of uniquely satisfiable 3SAT formulas. Is it super-polynomially dense or exponentially dense? What is known about the asymptotic lower bound on the number of 3SAT formulas with unique solutions?