Unavoidable Flats in Matroids Representable over Prime Fields

We show that, for any prime p and integer \(k \ge 2\), a simple \({{\,\textrm{GF}\,}}(p)\)-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 \({{\,\textrm{GF}\,}}(p)\)-representable matroids. For any prime p and integer \(k\ge 2\), if we 2-colour the elements in any simple \({{\,\textrm{GF}\,}}(p)\)-representable matroid with sufficiently high rank, then there is a monochromatic flat of rank k.