Excluding a line from $\mathbb C$-representable matroids

TL;DR

This work resolves a central extremal problem for simple matroids: among rank-$r$ matroids that are $ ext{C}$-representable and exclude a $U_{2,t+3}$-minor, the maximum size is $|M|=t{r\choose 2}+r$ for sufficiently large $r$, with equality exactly for rank-$r$ cyclic Dowling geometries of order $t$. The authors develop a unifying framework using group-labeled graphs to study $ ext{Γ}$-lift and $ ext{Γ}$-frame matroids, enabling a density-driven analysis that connects Dowling geometries, biased graphs, and tangles. They prove a strong main theorem (Theorem) that governs extremal behavior across minor-closed classes and derive exact and approximate corollaries for representability over various fields, as well as extensions to algebraic matroids and Δ-modular matrices. The results significantly deepen our understanding of how algebraic structure governs densest configurations in matroid theory and provide robust tools for identifying Dowling-like extremal objects. The methods have potential implications for related extremal problems in combinatorial geometry and linear-algebraic representations, offering a pathway to precise descriptions of densest matroids under broad representability constraints.

Abstract

For each positive integer $t$ and each sufficiently large integer $r$, we show that the maximum number of elements of a simple, rank-$r$, $\mathbb C$-representable matroid with no $U_{2,t+3}$-minor is $t{r\choose 2}+r$. We derive this as a consequence of a much more general result concerning matroids on group-labeled graphs.

Theorems & Definitions (264)

• Theorem 1.0.2: Kung
• Theorem 1.0.3: Geelen, Nelson
• Theorem 1.0.4
• Theorem 1.0.5
• Theorem 1.0.6
• Theorem 1.0.7
• Theorem 1.0.8: Growth Rate Theorem
• Theorem 1.0.9
• Theorem 2.1.1
• Theorem 2.1.2