Černý–Starke conjecture on synchronizing automata

A synchronizing automaton is a finite automaton whose transition semigroup contains a word sending every state to the same state; its reset threshold $rt(\mathcal{A})$ is the length of a shortest such synchronizing word. The Černý–Starke conjecture. Any synchronizing automaton on $n$ states has a synchronizing word of length at most

$$(n-1)^2.$$

This is a longstanding open problem in automata theory concerning the maximum reset threshold of synchronizing automata.