For the sake of getting this question off the unanswered stack, let me turn some of the comments into a question.

Noam Elkies' comment: if one considers arbitrary subsets of $S_n$, then one can find equations for which there are very large subsets of $S_n$ containing no solutions. For instance $x_1x_2x_3=1$ has no solutions in $S_n\setminus A_n$.

If one restricts to subsets of $A_n$, then one can exploit the fact that $A_n$ is perfect and so all non-trivial rep's have degree at least $2$. In fact, if $n\geq 7$, the minimal dimension of a non-trivial rep is $n-1$. Now the following paper is relevant:

*W. T. Gowers*, MR 2410393 **Quasirandom groups**, *Combin. Probab. Comput.* **17** (2008), no. 3, 363--387.

One of the main results in this paper gives conditions under which the equation $x_1x_2=x_3$ has solutions, and this theorem can be applied directly to $A_n$. The result in question is:

**Theorem 3.3**. Let $\Gamma$ be a finite group with no non-trivial representation of dimension less than $k$, let $n=|\Gamma|$ and let $A,B$ and $C$ be three subsets of $\Gamma$ such that $|A||B||C|>n^3/k$. Then there exist $a\in A, b\in B$ and $c\in C$ with $ab=c$. In particular, this is true if all of $A,B$ and $C$ have size greater than $n/k^{1/3}$.

The theorem goes on to give a lower bound for the number of solutions to the given equation. Generalizations of the theorem -- to equations involving more than three variables -- are discussed in Section 5 of the above paper.

Returning to the three variable situation, one should note that Kedlaya has proved a kind of converse to the above theorem, suggesting that the answer that Gowers gives for this equation is about as good as one could hope for. The paper of Kedlaya is:

K. S. Kedlaya, *Product-free subsets of groups,
then and now*, http://arXiv:0708.2295v1.

I don't know of any results concerning the other formulations of the OP's question -- where one has powers of variables in the equation.

Quasirandom groupsaddresses exactly this question of when sets are compelled to contain solutions to certain equations. He deals with three sets $A,B,C$ and the equation $a.b=c$. He confirms that a solution exists for all sets above a certain bound, which is a function of the order of the group and the minimal degree of a non-trivial complex representation. $\endgroup$ – Nick Gill Aug 2 '16 at 14:08arelarge subsets of $S_n$ with no solution of (say) $x_1 x_2 x_3 = e$. Either put some conditions on the exponents $a_k$ or restrict to subsets of $A_n$. $\endgroup$ – Noam D. Elkies Aug 2 '16 at 21:24