Lines imply spaces in density Ramsey theory

Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of Fⁿ there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see 9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in 3, 6] we obtain various consequences: a “group-theoretic” version of Roth's Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥F∥ = 2.