Extremal basis counts for binary Fano-minor-free matroids

Determine ex_N(n,r), the maximum number of bases of an n-element rank-r binary matroid with no F_7-minor, where F_7 is the Fano plane.

Background

The authors study whether the growth rate function of a minor-closed class can determine its extremal number of bases. For the class of binary matroids with no F_7-minor, Kung's result gives the growth rate h_N(r)=binom{r+1}{2}, attained by the graphic matroid of the complete graph on r+1 vertices.

Determining the corresponding maximum number of bases would generalize a theorem of Kelmans on maximizing the number of spanning trees in graphs with a fixed number of edges, provided extremal matroids can be taken to be graphic.

References

Let $n,r$ be integers with $n \ge r \ge 1$. Determine $\ex_{N}(n, r)$ for the class $N$ of binary matroids with no $F_7$-minor.

— Turán densities for matroid basis hypergraphs  (2502.03673 - Pol et al., 5 Feb 2025) in Section 4, subsection "Minor-closed classes"