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\}$ .

Similar problems for some different classes of groups are already answered:

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…