is a perturbation of the grid $G_d$. In [@LaSo07], the result was extended for pseudolines and homogeneous sets in $\R^n$ where pseudolines are a generalization of lines which include constant degree irreducible algebraic curves. Adding the homogeneous condition on a set is a much stronger condition (for sufficiently small $r$) than requiring that no large subset belongs to a hyperplane (however, we cannot derive these results from ours since our dependence on $d$ is suboptimal).

Overview of the proof
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The main tool used in the proof of Theorem \[thm-manyrichlines\] is a rank bound for design matrices. A [*design matrix*]{} is a matrix with entries in $\C$ and whose support (set of non-zero entries) forms a specific pattern. Namely, the supports of different columns have small intersections, the columns have large support and rows are sparse (see Definition \[def-designmatrix\]). Design matrices were introduced in  [@BDWY12; @DSW12] to study quantitative variants of the Sylvester-Gallai theorem. These works prove certain lower bounds on the rank of such matrices, depending only on the combinatorial properties of their support (see Section \[sec-design-matrices\]). Such rank bounds can be used to give [*upper bounds*]{} on the dimension of point configurations in which there are many ‘local’ 