Exact Tur7an basis numbers for binary matroids without an Fano-plane minor

Determine \(\ex_N(n,r)\), the maximum number of bases of an n-element, rank-r binary matroid with no F_7-minor, for all integers n r 1.

Background

The authors investigate whether the growth rate function of a minor-closed class can help determine the maximum number of bases in its matroids. As a concrete example, they consider the class N of binary matroids with no PG(2,2)-minor, equivalently no Fano-plane (F_7) minor.

Kung's result gives the growth rate of this class: the largest simple member of rank at most r has (r+12)\binom{r+1}{2} elements, with equality for the graphic matroid of the complete graph on r+1 vertices. The unresolved extremal question is to determine the corresponding maximum number of bases for arbitrary ground-set size n and rank r. A solution would generalize a result of Kelmans concerning spanning trees of graphs.

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 Problem, Section 7.4 (Minor-closed classes)

This leads to the following natural problem. Let $n,r$ be integers with $n r 1$. 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 "Directions for future work," subsection "Minor-closed classes," displayed problem following the discussion of the class N