TITLE: Special case of Erdos Distance Problem in a plane
QUESTION [8 upvotes]: Erdos in his Distinct distance Problem in a plane conjectured that the minimal number of distinct distance determined by $n$ points in a plane be $g(n)$,
$$g(n) \sim \frac{cn}{\sqrt{\log n}}$$
But for the special case which asks the minimum number of distinct distances that must be determined by $n$ points in a plane such that no $3$ are collinear be $g_{\textrm{no}3}(n)$, it is known that,
$$g_{\textrm{no}3}(n) \ge \left\lceil \frac{n-1}{3} \right\rceil$$
and $\displaystyle g_{\textrm{gen}}(n) \ge \left\lceil \frac{n-1}{3} \right\rceil$, where, $g_{\textrm{gen}}(n)$ asks the same question but for points in general position in a plane (i.e., no $3$ collinear and no $4$ conclyclic.)
Has there been any improvements on these particular casees of Erdos's Distance Problem?

REPLY [2 votes]: There has been recent progress on this problem published in the annals of mathematics:
On the Erdős distinct distances problem in the plane
Pages 155-190 from Volume 181 (2015), Issue 1 by Larry Guth, Nets Hawk Katz
"In this paper, we prove that a set of N points in $\mathbb{R}^2$ has at least $c\frac{N}{\log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erdős."