Erdos in his [Distinct distance Problem in a plane](https://en.wikipedia.org/wiki/Erd%C5%91s_distinct_distances_problem) 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?



Special case of Erdos Distance Problem in a plane
Here is a survey only four weeks old:



> 
> Sheffer, Adam. "Distinct distances: open problems and current bounds." arXiv:1406.1949 (2014). ([arXiv abstract](http://arxiv.org/abs/1406.1949v2)).
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It is not known whether $D\_{gen}(n)=\Theta(n)$
($\Omega(n)$ is trivial), and the current best
upper bound is $n 2^{O( \sqrt{\log n} )}$.


Adam also discusses $D\_{\mathop{no3}\ell}(n)$.
No nontrivial lower bound is known.


(Throughout, I believe Adam's $D\_{\ldots} =$ your $g\_{\ldots}$.)


