On what sets can $\mathfrak{S}_n$ act transitively? I would like to know $\mathfrak{S}_n$ could act faithfully transitively on sets with $m$ elements, with $m > n$. I know that it is not possible if $m = n+1$ except for $n = 5$. Any ideas ?

There are many sets of more than $n$ elements on which $\mathfrak S_n$ acts transitively. The largest possible example is that of the $n!$ total orderings of the set of $n$ elements (the one used to define $\mathfrak S_n$). One can deduce numerous smaller examples from this one.