  This article focuses on subset reachability in synchronizing automata. First,
we provide families of synchronizing automata with subsets which cannot be
reached with short words. These families do not fulfil Don's Conjecture about
subset reachability. Moreover, they show that some subsets need exponentially
long words to be reached, and that the restriction of the conjecture to
included subsets also does not hold. Second, we analyze completely reachable
automata and provide a counterexample to the conjecture of Bondar and Volkov
about the so-called $\Gamma_1$-graph. We finally prove an alternative version
of this conjecture.
