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

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

I have asked a similar question without forcing $X$ to generate, and received an example of an $X$, such that $|X| = (n - 5)!$, but $XXX \neq \langle X \rangle$. It was $S_{n-5} \times \{0; 1\}$, which lies in the subgroup $S_{n - 5} \times C_5$. However, it was conjectured in the comments, that the largest counterexamples, that generate $G$, are very likely to be much smaller.

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

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

I have asked a similar question without forcing $X$ to generate, and received an example of an $X$, such that $|X| = 2(n - 5)!$, but $XXX \neq \langle X \rangle$. It was $S_{n-5} \times \{0; 1\}$, which lies in the subgroup $S_{n - 5} \times C_5$. However, it was conjectured in the comments, that the largest counterexamples, that generate $G$, are very likely to be much smaller.

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

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

I have asked a similar question without forcing $X$ to generate, and received an example of an $X$, such that $|X| = (n - 5)!$, but $XXX \neq \langle X \rangle$. It was $S_{n-5} \times \{0; 1\}$, which lies in the subgroup $S_{n - 5} \times C_5$. However, it was conjectured in the comments, that the largest counterexamples, that generate $G$, are very likely to be much smaller.

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

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

I have asked a similar question without forcing $X$ to generate, and received an example of an $X$, such that $|X| = (n - 5)!$, but $XXX \neq \langle X \rangle$. It was $S_{n-5} \times \{0; 1\}$, which lies in the subgroup $S_{n - 5} \times C_5$. However, it was conjectured in the comments, that the largest counterexamples, that generate $G$, are very likely to be much smaller.