Tur\'an densities for matroid basis hypergraphs

TL;DR

This paper investigates the maximum number of bases in matroids avoiding a specific minor, linking it to Turán numbers of hypergraphs and providing solutions for particular cases.

Contribution

It connects matroid basis hypergraph extremal problems to Turán numbers and offers solutions for specific uniform minors.

Findings

Established a connection between matroid bases and Turán numbers.

Solved the maximum bases problem for certain uniform minors.

Extended Turán theory to matroid basis hypergraphs.

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 Tur\'an number of a daisy hypergraph to the family of matroid basis hypergraphs. We then answer this question for several interesting choices of $U$.