A density Hales–Jewett theorem for matroids

We show that if α is a positive real number, n and ℓ are integers exceeding 1, and q is a prime power, then every simple matroid M   of sufficiently large rank, with no U2,ℓU2,ℓ-minor, no rank-n   projective geometry minor over a larger field than GF(q)GF(q), and at least αqr(M)αqr(M) elements, has a rank-n   affine geometry restriction over GF(q)GF(q). This result can be viewed as an analogue of the multidimensional density Hales–Jewett theorem for matroids.