Determine the threshold for the star-candidate spectral supersaturation problem

Determine the threshold \(\delta_F^*\) for every color-critical graph \(F\) such that, for all sufficiently large \(n\) and integers \(q\leq \delta_F^* n\), the condition \(\lambda(G)\geq\lambda(T_{n,r,q})\), where \(T_{n,r,q}\) is obtained from \(T_{n,r}\) by adding a \(q\)-edge star inside a largest part, forces at least \(q\,c(n,F)\) copies of \(F\); in particular, determine whether the thresholds for cliques and odd cycles agree with their edge-setting thresholds.

Background

The matching-based candidate Yn,r,qY_{n,r,q} ceases to be extremal once qq exceeds the n\sqrt n scale. The paper notes that replacing the matching by a star permits a candidate graph Tn,r,qT_{n,r,q} for which the valid range of qq extends to linear order in nn.

The unresolved issue is to identify the optimal constant δF∗\delta_F^* for this star-based comparison graph. The authors further conjecture that, for cliques and odd cycles, the resulting thresholds should match the corresponding classical edge-setting thresholds.

References

It is interesting to determine the new threshold $\delta_F*$ for every color-critical graph $F$. We conjecture that the threshold $\delta_F*$ for cliques and odd cycles would be consistent with those in the edge setting; see Table \ref{tab:edge-vs-spec}.

— Optimal spectral supersaturation for cliques and odd cycles  (2610.06038 - Chen et al., 5 Oct 2026) in Section 6, Concluding remarks, paragraph beginning “The linear range of the increment”