Classification of Turan basis densities by growth-rate type

Determine whether the classification of extremal growth-rate functions for minor-closed classes of matroids induces a corresponding classification of their Turan basis densities.

Background

The Growth Rate Theorem gives four possible asymptotic behaviors for the growth-rate function of a minor-closed matroid class: linear, quadratic, exponential, or infinite.

The paper leaves unresolved whether these four structural regimes likewise determine qualitatively distinct behavior for the associated Turan basis densities.

References

We close with the following open-ended question: does the classification of extremal functions lead to a corresponding classification of Tur an basis densities $\pi_{M}(r)$ for minor-closed classes of matroids?

— Turán densities for matroid basis hypergraphs  (2502.03673 - Pol et al., 5 Feb 2025) in Final paragraph of Section 7.4, “Minor-closed classes”