Slowly synchronizing automata with fixed alphabet size

It was conjectured by Černý in 1964 that a synchronizing DFA on n states always has a shortest synchronizing word of length at most (n−1)2, and he gave a sequence of DFAs reaching this bound. In this paper, we investigate the role of the alphabet size. For each possible alphabet size, we count DFAs on n≤6 states which synchronize in (n−1)2−e steps, for all e3, we prove that the Černý automaton on n states does not admit non-trivial extensions with the same smallest synchronizing word length (n−1)2