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Excluding a line from positroids

Jonathan Boretsky, Zach Walsh

TL;DR

This work determines the exact maximum size of a simple rank-$r$ positroid that avoids a $U_{2,\ell+2}$-minor, proving $|M| \le \ell(r-1) + 1$ and showing equality precisely when the matroid is formed by parallel connections of copies of $U_{2,\ell+1}$. The authors integrate extremal matroid theory with structural properties of positroids and $M(K_4)$-minor-free classes, placing their bound among sharp results that generalize classical Kung-type problems. They also construct explicit extremal examples and propose several avenues for future research, including 3-connected positroids, other minor-closed classes, and oriented matroids, highlighting rich connections to geometry and physics.

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.

Excluding a line from positroids

TL;DR

This work determines the exact maximum size of a simple rank- positroid that avoids a -minor, proving and showing equality precisely when the matroid is formed by parallel connections of copies of . The authors integrate extremal matroid theory with structural properties of positroids and -minor-free classes, placing their bound among sharp results that generalize classical Kung-type problems. They also construct explicit extremal examples and propose several avenues for future research, including 3-connected positroids, other minor-closed classes, and oriented matroids, highlighting rich connections to geometry and physics.

Abstract

For all positive integers and , we determine the maximum number of elements of a simple rank- positroid without the rank- uniform matroid 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 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.

Paper Structure

This paper contains 11 sections, 8 theorems, 10 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids
  4. Positroids
  5. Excluded minors
  6. The extremal examples
  7. The main proof
  8. Future Work
  9. $3$-connected positroids
  10. Other $M(K_4)$-minor-free classes
  11. Oriented matroids

Key Result

Theorem 1.2

For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$. Moreover, if $r \ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.

Figures (4)

  • Figure 1: Lattice of flats of the uniform matroid $U_{2,\ell}$.
  • Figure 2: Some rank-$3$ excluded minors for the class of positroids.
  • Figure 3: A rank-$4$ excluded minor for the class of positroids.
  • Figure 4: The parallel connection of two matroids $M$ and $N$, each isomorphic to $U_{2,3}$.

Theorems & Definitions (30)

  • Theorem 1.2
  • Theorem 1.3: Geelen, Nelson Geelen-Nelson-2010
  • Theorem 1.4: Geelen, Nelson, Walsh Geelen-Nelson-Walsh-2024
  • Definition 2.1: Matroids in terms of flats
  • Remark 2.2
  • Example 2.3
  • Definition 2.4: Simple affine matroids
  • Example 2.5
  • Definition 2.6: Minors
  • Definition 2.7: Simple matroids
  • ...and 20 more