Černý’s conjecture on shortest synchronizing words

Prove that every synchronizable deterministic finite automaton (DFA) of order n has a synchronizing word whose length is at most (n−1)^2, thereby resolving Černý’s conjecture.

Background

The paper reviews the history and current status of Černý’s conjecture, which asserts a quadratic upper bound on the shortest reset word in any synchronizable DFA. Despite progress on special classes of automata and improvements to general cubic upper bounds, the conjecture remains unresolved. The authors’ cornering strategy provides synchronizing words for several structured DFAs and proves Černý’s conjecture under certain geometric conditions, but does not settle the conjecture in full generality.

This problem is foundational in automata theory and is referenced as a central difficulty motivating several parts of the paper, including their results for difference DFAs in Euclidean spaces and partially ordered DFAs.