Density of answers for NP complete problems? There is a result which goes along the lines of:


  Definition: For every language $L$, define $L_n = \{ w \in L \mid \text{length}(w) = n \}$, i.e. the subset of $L$ of words whose length is $n$.
  
  Theorem: If $\mathbf{P} = \mathbf{NP}$, then for every $\mathbf{NP}$-complete language $L$ there exists a polynomial $p : \mathbb{N} \rightarrow \mathbb{N}$ such that for every integer $n$ it is the case that $|L_n| \leq p(n)$.


The intuition of the theorem is that if $\mathbf{P} = \mathbf{NP}$, then $\mathbf{NP}$-complete languages have a polynomial density of solutions.

As we believe that $\mathbf{P} \neq \mathbf{NP}$, we should tend to believe the contrary, i.e. that $\mathbf{NP}$-complete problems do not have a polynomial density of solutions.

Example: For the case of 3-coloring, for large $n$ there are many 3-colorable graphs with $n$ nodes; in any case, the number of 3-colorable graphs is not bounded by a polynomial.

Does anyone recognise this result? What is the actual statement of the theorem? Any reference would be welcome!
You are defining what is known as sparse sets. Mahaney's theorem states that sparse $NP$-complete (under Karp reduction ) sets exist if and only if $P=NP$. So, your theorem is just one direction of that theorem.