If I have $n$ variables and I want to write down all 3-SAT problems, the number of problems is $2^{8{n \choose 3}}$, since each clause has 3 variables and each variable can be negated or not.

But empirically with SAT solvers, the hardest SAT problems have far fewer terms than this (if you have too few or too many terms SAT is easy to solve).

I suspect there is a subclass of 3-SAT that grows like $2^{(c_0+c_1 n)}$ and is still NP-complete.

Does anyone know of such a class or have suggestions on how to construct it?