The smallest matroids with no large independent flat

TL;DR

This paper characterizes the minimal simple rank-$r$ matroids lacking large independent flats, identifying a unique extremal structure for certain parameters.

Contribution

It introduces the matroid $M_{r,t}$ as the smallest example with no $(t+1)$-element independent flat and proves its uniqueness for $r \,\geq\, 2t$.

Findings

$M_{r,t}$ minimizes element count under the given conditions

Uniqueness of $M_{r,t}$ for $r \,\geq\, 2t$

Provides a characterization of extremal matroids with no large independent flat

Abstract

We show that a simple rank-$r$ matroid with no $(t+1)$-element independent flat has at least as many elements as the matroid $M_{r,t}$ defined as the direct sum of $t$ binary projective geometries whose ranks pairwise differ by at most $1$. We also show for $r \ge 2t$ that $M_{r,t}$ is the unique example for which equality holds.