It would be nice to find out what is known about the following problem.

First let us consider a free group $F$ with two generators $a$ and $b$. We are interested in its elements that are

- not equal to identity,
- of form $c_1 c_2 \ldots c_n d_1^{-1} d_2^{-1} \ldots d_n^{-1}$, where all $c_i$ and $d_i$ are either equal to $a$ or to $b$.

Let us denote all these elements by $W$.

Let us consider the group $S_k$ now. What is the shortest word from $W$ with the following property: whatever elements of $S_k$ we substitute for $a$ and $b$ we get identity (in $S_k$)?

The best bounds for the smallest $n$ I am aware of are $2^{O(k)}$ and $\Omega(k^2)$.