weaker version of Conjecture \[conj-rich\] in $\R^4$ trivially follows (see Appendix \[sec-stronger-thm1\]).

\[thm-conj-d=4\] Given a set $V$ of $n$ points in $\R^4$, let $s_2$ denote the maximum number of points of $V$ contained in a 2-flat and $s_3$ denote the maximum number of points of $V$ contained in a quadric hypersurface or a hyperplane. Then there is an absolute constant $c>0$ such that for $r\ge 2$, $$|\L_r(V)|\lesssim     2^{c\sqrt{\log n}}\cdot \left(\frac{n^2}{r^5}+\frac{ns_3}{r^4}+\frac{ns_2}{r^3}+\frac{n}{r}\right).$$

We are not aware of any examples where points arranged on a quadric hypersurface in $\R^4$ result in significantly more rich lines than in a four dimensional grid. It is, however, possible that one needs to weaken Conjecture \[conj-rich\] so that for some $1<\ell<d$, an $\ell$-dimensional hypersurface of constant degree (possibly depending on $\ell$) contains $\gtrsim_d n/r^{d-\ell}$ points.

To make the comparison with the above theorems easier, Theorem \[thm-manyrichlines\] can be stated equivalently as follows:

\[thm-manyrichlines-equiv\] Given a set $V$ of $n$ points in $\C^d$, let $s_{d-1}$ denote the maximum number of points of $V$ contained in a hyperplane. Then for $r\ge 2$, $$|\L_r(V)| \lesssim_d \frac{n^2}{r^d} + \frac{ns_{d-1}}{r^2}.$$

In [@SoVu04], it was shown that $|\L_r(V)|\lesssim_d \frac{n^2}{r^{d+1}}$ when $V\subset \R^d$ is a [*homogeneous*]{} set. This roughly means that the point set 