  We investigate the length $\ell(n,k)$ of a shortest preset distinguishing
sequence (PDS) in the worst case for a $k$-element subset of an $n$-state Mealy
automaton. It was mentioned by Sokolovskii that this problem is closely related
to the problem of finding the maximal subsemigroup diameter
$\ell(\mathbf{T}_n)$ for the full transformation semigroup $\mathbf{T}_n$ of an
$n$-element set. We prove that
$\ell(\mathbf{T}_n)=2^n\exp\{\sqrt{\frac{n}{2}\ln n}(1+ o(1))\}$ as
$n\to\infty$ and, using approach of Sokolovskii, find the asymptotics of
$\log_2 \ell(n,k)$ as $n,k\to\infty$ and $k/n\to a\in (0,1)$.
