Characterize supersaturation in the intermediate range above the matching threshold

Characterize the extremal candidate graph and determine the correct order of \(\#F(G)\) under the condition \(\lambda(G)\geq\lambda(Y_{n,r,q})\) in the intermediate regime \(\sqrt n\ll q\ll n\), where the matching-based graph \(Y_{n,r,q}\) is no longer extremal.

Background

The main theorems establish sharp results only for qq below a constant multiple of n\sqrt n. Above the relevant threshold, the paper gives constructions showing that the matching-based comparison graph no longer forces the same supersaturation count.

The interval between the n\sqrt n and linear scales is explicitly identified as unresolved. Solving this problem requires finding the appropriate replacement for Yn,r,qY_{n,r,q} and determining the resulting copy-count asymptotics.

References

It would be interesting to describe the behavior above the threshold, say $q \gg \sqrt n$, where the candidate graph $Y_{n,r,q}$ is no longer extremal, and the intermediate range $\sqrt n\ll q\ll n$ is widely open. Since the condition $\lambda(G)\geq\lambda(Y_{n,r,q})$ no longer forces $#F(G) \geq q\,c(n,F)$ in this range, it would be worth determining the new candidate graph and the correct order of $#F(G)$.

— 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”