Excluding a line from positroids

TL;DR

This paper determines the maximum size of simple rank-$r$ positroids avoiding a specific minor, extending bounds known for other matroid classes and providing methods for future research in related structures.

Contribution

It establishes the maximum number of elements in such positroids and characterizes those achieving this maximum, pioneering the study of positroids in this context.

Findings

Maximum element count for simple rank-$r$ positroids without $U_{2, ext{ell}+2}$ minor.

Characterization of extremal positroids with maximum elements.

Methodology for analyzing minor-exclusion in positroids.

Abstract

For all positive integers $\ell$ and $r$, we determine the maximum number of elements of a simple rank-$r$ positroid without the rank-$2$ uniform matroid $U_{2,\ell+2}$ as a minor, and characterize the matroids with the maximum number of elements. This result continues a long line of research into upper bounds on the number of elements of matroids from various classes that forbid $U_{2,\ell+2}$ as a minor. This is the first paper to study positroids in this context, and it suggests methods to study similar problems for other classes of matroids, such as gammoids or base-orderable matroids.