## Is there some sort of formula for $\tau(S_n)$?

Let $$G$$ be a finite group. Define $$\tau(G)$$ as the minimal number, such that $$\forall X \subset G$$ if $$|X| > \tau(G)$$, then $$XXX = \langle X \rangle$$. Is there some sort of formula for $$\tau(S_n)$$, for the symmetric group $$S_n$$?

Here $$XXX$$ stands for $$\{abc| a, b, c \in X\}$$.

1) $$\tau(\mathbb{C}_n) = \lceil \frac{n}{3} \rceil + 1$$, where $$\mathbb{C}_n$$ is cyclic of order $$n$$;
2) Gowers, Nikolov and Pyber proved the fact that $$\tau(\mathrm{SL}(n, p)) \leq 2|\mathrm{SL}(n, p)|^{1-\frac{1}{3(n+1)}}$$ for prime $$p$$.
However, I have never seen anything like that for $$S_n$$. It will be interesting to know if there is something…