# Tur\'an's theorem for Dowling geometries

## TL;DR

This paper extends Turán's theorem to Dowling geometries, determining maximum sizes of N-free submatroids and revealing complex dependencies on the underlying group structure.

## Contribution

It generalizes Turán's theorem to Dowling geometries, analyzing maximum N-free submatroids for various N and group structures, including nontrivial groups.

## Key findings

- Maximum size of N-free submatroids determined for various N
- Complex dependence on the group mma when mma is nontrivial
- Reduction to classical Ture1n's theorem when mma is trivial

## Abstract

The Dowling geometry $Q_n(\Gamma)$, where $\Gamma$ is a finite group, is a matroid that generalizes the complete-graphic matroid $M(K_{n+1})$. We determine the maximum size of an $N$-free submatroid of $Q_n(\Gamma)$ for various choices of $N$, including subgeometries $Q_m(\Gamma')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $\Gamma$ is trivial and $N=M(K_t)$, this problem reduces to Tur\'{a}n's classical result in extremal graph theory. We show that when $\Gamma$ is nontrivial, a complex dependence on $\Gamma$ emerges, even when $N=M(K_4)$.