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

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 $t\left(\genfrac{}{}{0ex}{}{r}{2}\right)+r$. We derive this as a consequence of a much more general result concerning matroids on group-labeled graphs.