Fukuda–Ziegler question on g(n)

The Fukuda–Ziegler question asks for the asymptotic behavior of the maximum facet count $g(n)$ for full-dimensional $0/1$-polytopes. The retrieved historical source does not specify when the question was posed.

Known results

• Fleiner–Kaibel–Rote: $g(n)\leq 30(n-2)!$ for sufficiently large $n$.
• Bárány–Pór: $g(n)\geq\left(\frac{cn}{\log n}\right)^{n/4}$.
• Gatzouras–Giannopoulos–Markoulakis (2004): $g(n)\geq\left(\frac{cn}{\log^{2}n}\right)^{n/2}$.

August 2026 near-factorial lower bound

A preprint claims the improved bound $g(n)\geq(cn)^{n/2}$ for sufficiently large $n$, removing the logarithmic loss and determining $\log g(n)$ up to an $O((\log n)^2)$ error when combined with the factorial upper bound. It does not determine an exact formula. A related preprint discloses extensive ChatGPT assistance in exploring and refining the proof, while stating that the author independently verified the arguments; the mathematical claim remains unverified.

Current status (as of August 2026): The factorial upper bound and several exponential lower bounds are known, while the new near-factorial lower bound is claimed but unverified and the exact behavior of $g(n)$ remains open.