Unavoidable flats in matroids representable over prime fields

TL;DR

This paper proves that high-rank GF(p)-representable matroids necessarily contain specific flats, leading to a Ramsey-type theorem ensuring monochromatic flats in 2-coloured matroids.

Contribution

It establishes the existence of particular flats in high-rank GF(p)-representable matroids and derives a new Ramsey-type result for such matroids.

Findings

High-rank GF(p)-representable matroids contain flats that are either independent or geometries.

A Ramsey-type theorem guarantees monochromatic flats in 2-colourings of high-rank GF(p)-representable matroids.

The results unify structural properties and colouring principles in matroid theory.

Abstract

We show that, for any prime $p$ and integer $k \geq 2$, a simple GF($p$)-representable matroid with sufficiently high rank has a rank-$k$ flat which is either independent in $M$, or is a projective or affine geometry. As a corollary we obtain a Ramsey-type theorem for GF($p$)-representable matroids. For any prime $p$ and integer $k\ge 2$, if we $2$-colour the elements in any simple GF($p$)-representable matroid with sufficiently high rank, then there is a monochromatic flat with rank $k$.