Optimize the threshold for outerplanar extremality

Optimize the threshold \(f_1\) in the outerplanar extremal theorem and extract the smallest possible explicit threshold beyond which the fan \(k\)-angulation is uniquely extremal.

Background

Theorem 2 proves that the fan is uniquely extremal once the number of faces exceeds a threshold f1f_1, but the proof does not optimize this threshold. The authors note that their estimates permit an explicit threshold to be extracted, leaving its quantitative improvement unresolved.

References

Three quantitative points remain open in the outerplanar case. The first is the exponent \varrho_k of Theorem~\ref{thm:stab}, which we have not tried to optimize; it is what balances the two error terms of the master inequality, and it is far from what the extremal hypergraph suggests. Because Section~\ref{sec:hub} uses only the qualitative content of Theorem~\ref{thm:stab}, improved quantitative bounds in Theorem~\ref{thm:stab} could reduce the threshold f_1 of Theorem~\ref{thm:main}; the estimates allow an explicit threshold to be extracted, but we have not optimized it.

— The maximum spectral radius of outerplanar and planar $k$-uniform hypergraphs  (2609.10660 - Liu et al., 9 Sep 2026) in Section 6, Concluding Remarks

Whether the fan k-angulation is extremal for every admissible n, with finitely many exceptions listed explicitly, is open for every k\ge3.

— The maximum spectral radius of outerplanar and planar $k$-uniform hypergraphs  (2609.10660 - Liu et al., 9 Sep 2026) in Section 6, Concluding Remarks