Let $S_n$ be the permutation group on $n$ elements. Denote by $K(n)$ the largest $k$ s.t. $S_n$ has a $k$-transitive subgroup (w.r.t. its action on the $n$-element set on which $S_n$ acts) different from $S_n,A_n$.

I heard that it has been proved that $K(n)\le 7$ for all $n$ but the proof uses the classification of finite groups (please correct me if I am wrong on this). But I am also interested in the exact value of $K(n)$ for small $n$ (or even all $n$ if there is a simple description).

Also I wonder what can be proved in a more elementary way (even weak bounds like $K(n)<n/3$ are interesting).