Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Girth in $GF(q)$-representable matroids

James Davies, Meike Hatzel, Kolja Knauer, Rose McCarty, Torsten Ueckerdt

TL;DR

The paper resolves the Geelen–Gerards–Whittle conjecture on girth in $GF(q)$-representable matroids by proving that for each $t$ and $q$ there exists a function $f(t,q)$ such that every cosimple $GF(q)$-representable matroid with girth at least $f(t,q)$ contains either $M(K_t)$ or $M(K_t)^*$ as a minor. The proof combines the Growth Rate Theorem for GF($q$)-representable matroids with Haussler's Shallow Packing Lemma, encoding the matroid via a representation $[I|A]$ to define a set-system whose shatter function is tightly controlled. A key dichotomy shows that either the set-system is not $oldsymbol{oldsymbol}$-separated or its size is bounded, and in the former case a small circuit leads to the required minor bound $k=oldsymbol{oldsymbol}+1$, establishing the result. This extends classical graph-minor phenomena (Thomassen, Mader) to the matroid setting and points to a broader program of classifying unavoidable cosimple matroids of large girth via minor-minimal families, with potential generalizations beyond GF($q$)-representability.

Abstract

We prove a conjecture of Geelen, Gerards, and Whittle that for any finite field $GF(q)$ and any integer $t$, every cosimple $GF(q)$-representable matroid with sufficiently large girth contains either $M(K_t)$ or $M(K_t)^*$ as a minor.

Girth in $GF(q)$-representable matroids

TL;DR

The paper resolves the Geelen–Gerards–Whittle conjecture on girth in -representable matroids by proving that for each and there exists a function such that every cosimple -representable matroid with girth at least contains either or as a minor. The proof combines the Growth Rate Theorem for GF()-representable matroids with Haussler's Shallow Packing Lemma, encoding the matroid via a representation to define a set-system whose shatter function is tightly controlled. A key dichotomy shows that either the set-system is not -separated or its size is bounded, and in the former case a small circuit leads to the required minor bound , establishing the result. This extends classical graph-minor phenomena (Thomassen, Mader) to the matroid setting and points to a broader program of classifying unavoidable cosimple matroids of large girth via minor-minimal families, with potential generalizations beyond GF()-representability.

Abstract

We prove a conjecture of Geelen, Gerards, and Whittle that for any finite field and any integer , every cosimple -representable matroid with sufficiently large girth contains either or as a minor.
Paper Structure (3 sections, 5 theorems, 1 equation, 1 figure)

This paper contains 3 sections, 5 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. The proof
  3. Conclusion

Key Result

Theorem 1

For any finite field $\textsf{GF}(q)$ and any integer $t$, there exists an integer $f(t,q)$ such that every cosimple $\textsf{GF}(q)$-representable matroid with girth at least $f(t,q)$ contains either $M(K_t)$ or $M(K_t)^*$ as a minor.

Figures (1)

  • Figure 1: A ternary matroid $M$ with a representation $[ I\text{ | }A ]$ over $\textsf{GF}(3)$ and the corresponding set system $\mathcal{F}$ with $e \in E(M)\setminus B$ and $F_e = \{(b_1,1), (b_2,1)\}$ highlighted in blue.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2: geelen2003cliques
  • Lemma 3: Haussler95
  • Theorem 4
  • proof
  • Claim 1
  • proof
  • Theorem 5: geelen2003cliques
  • Conjecture 6
  • Conjecture 7
  • ...and 1 more