I asked Jon Lenchner, an expert on point-line incidences, and he told me the
question (in dual form) was posed in
Grünbaum's 1971 book
_[Arrangements and Spreads][1]_,
and fully answered in a paper by
Peter Salamon and
Paul Erdős:
"The solution to a problem of Grünbaum,"
_Canad. Math. Bull._ <b>31</b>: 129-138 (1988).
Here are its first two sentences:
> In the paper below we characterize for large $n$ the possible values of the
number of connecting lines determined by a set of $P_n$ points in the plane,
where a connecting line is any straight line containing at least two points of $P_n$.
This solves a problem posed by Grünbaum [5,6] which asks for the sequence
of all integers $m$ with the property that some configuration of $n$
points determine exactly $m$ lines.
([6] is _Arrangements and Spreads_; [5] is Erdős's earlier partial solution.)
They obtain exact expressions "for the lower end of the continuum of values leading
down from $\binom{n}{2}-4$." "The possible values...can be seen to bear a strong resemblance
to physical spectra."
The lower end of the continuum grows as $n^{3/2}$ (with constant 1).
Here are two figures from the paper:
<br />
![SE Fig1][2]<br />
![SE Fig4][3]
[1]: http://books.google.com/books/about/Arrangements_and_spreads.html?id=2L5Mgc-6vN8C
[2]: https://i.sstatic.net/HhQc2.jpg
[3]: https://i.sstatic.net/y3AYP.jpg