Let $G$ be a group with two generators. Suppose that all non-trivial words of length less or equal $n$ in the generators and their inverses define non-trivial elements in $G$.

Question:How many of the $4\cdot 3^{n}$ words of length $n+1$ in the generators and their inverses can at most be trivial in $G$?

I am interested in the growth of this number as $n$ grows. From what I understand from the Gromov's theory of random groups, a choice of relations of length $n+1$ will (almost surely as $n \to \infty$) not enforce shorter relations if one chooses $$3^{\left(\frac{1}2 - \varepsilon\right) \cdot n}$$ relations of length $n+1$ at random. (This result is related to small cancellation theory which applies to randomly choosen relations. The exponent $1/2$ which appears is related to the birthday paradox. It ensures that with high probability one does not chose relations which have large overlap.) However, I would not know how to prove it or even locate it in the literature. Can someone confirm this?

Question:Can one do better than $3^{\left(\frac{1}2 - \varepsilon\right)\cdot n}$ (as $n \to \infty$) with a concrete sequence of groups rather than using random groups?