Turán's theorem for Dowling geometries

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 $Q_m({\Gamma }^{\prime })$, lines $U_{2,\ell }$, and graphic matroids $M(H)$. When the group $\Gamma$ is trivial and $N=M(K_t)$, this problem reduces to Turán's classical result in extremal graph theory. We show that when $\Gamma$ is nontrivial, a complex dependence on $\Gamma$ emerges, even when $N=M(K_4)$.

18 pages; in this version, minor improvements were made to exposition and mathematics