The number of rank-$k$ flats in a matroid with no $U_{2,n}$-minor

TL;DR

This paper establishes an upper bound on the number of rank-$k$ flats in large rank matroids without a certain minor, relating it to projective geometries over finite fields, thus extending matroid extremal theory.

Contribution

It proves a new extremal bound for the number of flats in matroids avoiding a specific minor, connecting matroid structure to finite field geometries.

Findings

Upper bound matches flats in projective geometries over GF(q)

Valid for sufficiently large rank r

Extends extremal matroid theory

Abstract

We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no $U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a rank-$r$ projective geometry over GF$(q)$, where $q$ is the largest prime power not exceeding $\ell$.