Turán densities for matroid basis hypergraphs
Jorn van der Pol, Zach Walsh, Michael C. Wigal
TL;DR
This work reframes the problem of maximizing the number of bases in $n$-element, rank-$r$ matroids under minor-avoidance constraints as Turán-type questions on matroid basis hypergraphs. The authors establish the existence and compute several Turán densities $\pi_{\mathsf M}(r,N)$ for uniform minors $N=U_{s,t}$, deriving sharp upper bounds and infinite-series limits, notably showing $\pi_{\mathsf M}(r, U_{2,t+2}) = \prod_{i=1}^{r-1}(1 - t^{-i})$ and $\lim_{r\to\infty}\pi_{ \text{M}}(r,U_{2,t+2}) = \prod_{i\ge1}(1 - t^{-i})$. They also determine exact densities for several cases with small $s$ or special $t$, such as $\pi_{ \text{M}}(3,U_{3,4}) = 4/9$ and $\pi_{ \text{M}}(3,U_{3,5}) = 3/4$, connecting matroid Turán densities to classical hypergraph results via daisy constructions and projective-geometric methods. Beyond specific densities, the paper develops a robust framework linking matroid minor-free problems to hypergraph Turán theory through suspensions and forbidden induced subgraphs, and highlights structural phenomena in rank-3 matroids with restricted minors, offering a range of directions for future work in discrete geometry and extremal matroid theory.
Abstract
Let $U$ be a uniform matroid. For all positive integers $n$ and $r$ with $n \ge r$, what is the maximum number of bases of an $n$-element, rank-$r$ matroid without $U$ as a minor? We show that this question arises by restricting the problem of determining the Turán number of a daisy hypergraph to the family of matroid basis hypergraphs. We then answer this question for several interesting choices of $U$.