Determine the sharp single-face increment of the fan functional

Determine the asymptotic sharp lower bound, or exact asymptotic behavior, of the single-face increment \(\theta_{d+1}-\theta_d\) for the fan functional as \(d\to\infty\).

Background

The fan functional has asymptotic size θd=ckd1/k(1+o(1))\theta_d=c_kd^{1/k}(1+o(1)), but the authors explain that this only yields useful increment information when faces are added in blocks. Their proof uses a weaker one-face increment bound from the gluing argument; a sharp estimate would simplify the comparison used to prove outerplanar extremality.

References

The second is the increment of the fan functional. The gluing argument of Lemma~\ref{lem:super} gives a lower bound with constant \gamma_k=c_k/(4k). As Remark~\ref{rem:increment} notes, the asymptotic \theta_d=c_kd{1/k}(1+o(1)) yields increments only in blocks of faces and does not by itself determine the asymptotic of the single-face increment. The lower bound suffices here, but a sharp increment estimate would simplify the comparison in Section~\ref{sec:hub}.

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