n/r^{d-\ell}$ each. Each of these grids will have $\gtrsim_d \alpha^2n^2/r^{2d-\ell+1}$ $r$-rich lines and so, together we will get $\gtrsim_d \alpha n^2/r^{d+1}$ rich lines. We can also take a union of $n/r$ lines containing $r$ points each, to get more $r$-rich lines than in the $d$-dimensional grid $G_d$ when $r\gg_d n^{1/d}$. We thus arrive at the following conjecture which, if true, would mean that the best one can do is to paste together a number of grids as above.

\[conj-rich\] For $r\ge 2$, suppose $V \subset \C^d$ is a set of $n$ points with $$|\cL_r(V)| \gg_d \frac{n^2}{r^{d+1}}+\frac{n}{r}.$$ Then there exists $1<\ell<d$ and a subset $V' \subset V$ of size $ \gtrsim_d n/r^{d-\ell}$ which is contained in an $\ell$-flat (i.e., an $\ell$-dimensional affine subspace).

This conjecture holds in $\R^3$ [@GK10] and, in a slightly weaker form, in $\R^4$ [@SS14]. We compare these two results with ours later in the introduction. Our main result makes a step in the direction of this conjecture. First of all, our bound is off by a factor of $r$ from the optimal bound (i.e., with $n^2/r^{d}$ instead of $n^2/r^{d+1}$). Secondly, we are only able to detect a $d-1$ dimensional subset (instead of finding the correct $\ell$ which 