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Existence of any quadratic upper bound on the Černý function

Determine whether there exists a universal upper bound that is quadratic in n for the Černý function 𝔠(n), the maximum reset threshold among all n‑state synchronizing deterministic finite automata.

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Background

Beyond the exact conjectured value 𝔠(n)=(n−1)2, the authors highlight a broader unresolved issue: whether any quadratic upper bound on 𝔠(n) exists. The best known bounds are cubic (Pin–Frankl, later slightly improved by Szykuła and Shitov), leaving open even the existence of an O(n2) bound.

This question is independent of the full Černý conjecture and would already represent a significant breakthrough if answered affirmatively.

References

As of August 2025, not only the truth of the equality \mathfrak{C}(n)=(n-1)2, but also the existence of any upper bound on \mathfrak{C}(n) that is quadratic in n, remain unknown.

— List of Results on the Černý Conjecture and Reset Thresholds for Synchronizing Automata (2508.15655 - Volkov, 21 Aug 2025) in Subsection 2.2, “Synchronizing automata and the Černý conjecture”