Resolve the Černý conjecture

Establish whether the reset threshold of every synchronizing deterministic finite automaton with n states is at most (n−1)^2; equivalently, prove or refute that the Černý function 𝔠(n), defined as the maximum reset threshold over all n‑state synchronizing deterministic finite automata, satisfies 𝔠(n)=(n−1)^2 for all n.

Background

The paper defines the Černý function 𝔠(n) as the maximum reset threshold among all synchronizing automata with n states and recalls the classical Černý conjecture asserting 𝔠(n)=(n−1)^2. Despite decades of work and many partial results for special classes of automata, the general conjecture has resisted resolution.

The authors note the current status as of August 2025 and provide historical context, including known lower bounds from Černý’s series and best known general upper bounds (which are cubic), underscoring that the conjecture itself is still unsettled.