Conjectured direct-sum extremizers for circumference-bounded matroids

Prove that, for s>=4 and r>=s-1, the maximum number of bases in an n-element, rank-r matroid with no U_{s,s+1}-minor is at most the stated direct-sum bound, with equality under the divisibility conditions s-1|r and r/(s-1)|n for a direct sum of r/(s-1) copies of U_{s-1,n/(r/(s-1))}.

Background

The paper determines the extremal basis densities for exclusions U_{2,3} and U_{3,4}, where direct sums of uniform matroids of rank s-1 provide the extremal constructions.

For s>=4, the authors expect the same pattern but do not have the structural argument needed to prove it. The conjecture proposes the exact finite Turan basis number and its equality cases.

References

We next consider $s \ge 4$. Matroids with circumference $s$ are not as well-structured when $s \ge 4$, but we expect that $\ex_{M}(n,r,U_{s,s+1})$ is achieved by a direct sum of uniform matroids of rank $s - 1$, as it is for $s = 2$ and $s = 3$. \begin{conjecture} Let $s,r$ be positive integers with $s \ge 4$ and $r \ge s - 1$. Then $\ex_{M}(n,r,U_{s,s+1}) \le \binom{\frac{n}{r/(s-1)}{s-1}{\frac{r}{s-1}$ for all $n \ge r$, with equality when $s - 1$ divides $r$ and $\frac{r}{s-1}$ divides $n$ for the direct sum of $r/(s-1)$ copies of $U_{s-1, \frac{n}{r/(s-1)}$. \end{conjecture}

— Turán densities for matroid basis hypergraphs  (2502.03673 - Pol et al., 5 Feb 2025) in Conjecture in Section 5, “U_{s,s+1}”