@article{Brown1982ADV,
title={A Density Version of a Geometric Ramsey Theorem},
author={Tom C. Brown and Joe Buhler},
journal={J. Comb. Theory, Ser. A},
year={1982},
volume={32},
pages={20-34}
}

Let V be an n-dimensional affine space over the field with pd elements, p 6= 2. Then for every ε > 0 there is an n(ε) such that if n = dim(V ) n(ε) then any subset of V with more than εjV j elements must contain 3 collinear points (i.e., 3 points lying in a one-dimensional affine subspace).