Optimize the stability exponent

Optimize the exponent \(\varrho_k\) in the outerplanar stability theorem and determine quantitatively sharper bounds than those currently obtained for extremal \(k\)-angulation hypergraphs.

Background

Theorem 2 establishes outerplanar stability with the exponent ϱk=1/[k(3k+1)]\varrho_k=1/[k(3k+1)]. The authors state that this exponent was chosen to balance error terms rather than optimized, and that it is far from the behavior suggested by the extremal hypergraph. Sharper quantitative stability estimates could also lower the threshold required for the eventual extremal characterization.

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.

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