Representability of matroids with a large projective geometry minor

Abstract

We prove that for each prime power $q$ there is an integer $n$ such that if $M$ is a $3$-connected, representable matroid with a PG$(n-1,q)$-minor and no $U_{2,q^2+1}$-minor, then $M$ is representable over GF$(q)$. We also show that for $\ell >=2$, if $M$ is a $3$-connected, representable matroid of sufficiently high rank with no $U_{2,\ell +2}$-minor and $|E(M)|\ge (4\ell {)}^{r(M)/2}$, then $M$ is representable over a field of order at most $\ell$.

10 pages

References in corpus (1)

• Confinement of matroid representations to subsets of partial fields