Unavoidable flats in matroids representable over prime fields

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 $k$.