Representability of matroids with a large projective geometry minor

TL;DR

This paper proves that large, highly connected, representable matroids with certain minors are representable over specific finite fields, establishing conditions linking minors, connectivity, and field representability.

Contribution

It establishes new conditions under which matroids with large projective geometry minors are representable over particular finite fields.

Findings

Matroids with large PG minors are GF(q)-representable under certain connectivity and minor exclusion conditions.

High-rank matroids with no small U_{2,ell+2}-minor are representable over fields of bounded size.

Explicit bounds relate matroid size, rank, and field representability.

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 $|E(M)| \geq (4\ell)^{r(M)/2}$, then $M$ is representable over a field of order at most $\ell$.