  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 $k$.
